formulasearchengine - User contributions [en]

No free lunch in search and optimization

2015-01-07T11:26:58Z

193.190.253.145: changed pays to might pay. The prices are randomly shuffled, and by saying pays we are implying that the price of vegan is always inversely higher when compared to carnivore.

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== am facing the mess left by the previous captain daze. ==

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== can not let the brother ==

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== for example ==

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Pion

2013-12-23T10:06:23Z

193.190.253.150:

{{redirect|Delta function|other uses|Delta function (disambiguation)}}
[[Image:Dirac distribution PDF.svg|325px|thumb|Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.]]
[[Image:Dirac function approximation.gif|right|frame|The Dirac delta function as the limit (in the sense of [[distribution (mathematics)|distributions]]) of the sequence of zero-centered [[normal distribution]]s <math>\delta_a(x) = \frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2}</math> {{nowrap|1=as <math>a \rightarrow 0</math>.}}]]

In mathematics, the '''Dirac delta function''', or '''{{mvar|δ}} function''', is (informally) a [[generalized function]] on the real number line that is zero everywhere except at zero, with an [[integral]] of one over the entire real line.<ref name=Dirac1958p58>{{harvnb|Dirac|1958|loc=§15 The δ function}}, p. 58</ref><ref>{{harvnb|Gel'fand|Shilov|1968|loc=Volume I, §§1.1, 1.3}}</ref><ref>{{harvnb|Schwartz|1950|p=3}}</ref> The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized [[point mass]] or [[point charge]].<ref>{{harvnb|Arfken|Weber|2000|p=84}}</ref> It was introduced by theoretical physicist [[Paul Dirac]]. In the context of [[signal processing]] it is often referred to as the '''unit impulse symbol''' (or function).<ref name="Bracewell 1986 loc=Chapter 5">{{harvnb|Bracewell|1986|loc=Chapter 5}}</ref> Its discrete analog is the [[Kronecker delta]] function which is usually defined on a finite domain and takes values 0 and 1.

From a purely mathematical viewpoint, the Dirac delta is not strictly a [[function (mathematics)|function]], because any extended-real function that is equal to zero everywhere but a single point must have total integral zero.<ref>{{harvnb|Vladimirov|1971|loc=§5.1}}</ref> The delta function only makes sense as a mathematical object when it appears inside an integral. While from this perspective the Dirac delta can usually be manipulated as though it were a function, formally it must be defined as a [[Distribution (mathematics)|distribution]] that is also a [[Measure (mathematics)|measure]]. In many applications, the Dirac delta is regarded as a kind of limit (a [[weak limit]]) of a [[sequence]] of functions having a tall spike at the origin. The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

==Overview==
The [[graph of a function|graph]] of the delta function is usually thought of as following the whole ''x''-axis and the positive ''y''-axis. Despite its name, the delta function is not truly a function, at least not a usual one with range in [[real number]]s. For example, the objects ''f''(''x'') = δ(''x'') and ''g''(''x'') = 0 are equal everywhere except at ''x'' = 0 yet have integrals that are different. According to [[Lebesgue integral#Basic theorems of the Lebesgue integral|Lebesgue integration theory]], if ''f'' and ''g'' are functions such that ''f'' = ''g'' [[almost everywhere]], then ''f'' is integrable [[if and only if]] ''g'' is integrable and the integrals of ''f'' and ''g'' are identical. Rigorous treatment of the Dirac delta requires [[measure theory]] or the theory of [[distribution (mathematics)|distribution]]s.

The Dirac delta is used to model a tall narrow spike function (an ''impulse''), and other similar [[abstraction]]s such as a point [[electric charge|charge]], point [[mass]] or [[electron]] point. For example, to calculate the [[dynamics (mechanics)|dynamics]] of a [[baseball]] being hit by a bat, one can approximate the [[force]] of the bat hitting the baseball by a delta function. In doing so, one not only simplifies the equations, but one also is able to calculate the [[motion (physics)|motion]] of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball.

In applied mathematics, the delta function is often manipulated as a kind of limit (a [[weak limit]]) of a [[sequence]] of functions, each member of which has a tall spike at the origin: for example, a sequence of [[Gaussian distribution]]s centered at the origin with [[variance]] tending to zero.

==History==
[[Joseph Fourier]] presented what is now called the [[Fourier integral theorem]] in his treatise ''Théorie analytique de la chaleur'' in the form:<ref name=Fourier>{{cite book |title=The Analytical Theory of Heat |url=http://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA408&dq=%22when+the+integrals+are+taken+between+infinite+limits%22+%22that+is+to+say,+that+we+have+the+equation%22&hl=en&sa=X&ei=rFe-T96cEIzKiQKcyfDtDQ&ved=0CD4Q6AEwAA#v=onepage&q=%22when%20the%20integrals%20are%20taken%20between%20infinite%20limits%22%20%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&f=false |author=JB Fourier |year=1822 |page=408 |edition= English translation by Alexander Freeman, 1878 |publisher=The University Press}} The original French text can be found [http://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA525&dq=%22c%27est-%C3%A0-dire+qu%27on+a+l%27%C3%A9quation%22&hl=en&sa=X&ei=SrC7T9yKBorYiALVnc2oDg&sqi=2&ved=0CEAQ6AEwAg#v=onepage&q=%22c%27est-%C3%A0-dire%20qu%27on%20a%20l%27%C3%A9quation%22&f=false here].</ref>

:<math>f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty\ \ d\alpha f(\alpha) \ \int_{-\infty}^\infty dp\ \cos (px-p\alpha)\ , </math>

which is tantamount to the introduction of the δ-function in the form:<ref name= Kawai>{{cite book |title=Microlocal Analysis and Complex Fourier Analysis |editor=Takahiro Kawai, Keiko Fujita, eds|author=Hikosaburo Komatsu |chapter=Fourier's hyperfunctions and Heaviside's pseudodifferential operators |isbn=9812381619 |year=2002 |publisher=World Scientific |url=http://books.google.com/books?id=8GwKzEemrIcC&pg=PA200&dq=%22Fourier+introduced+the%22+%22+-function+much+earlier%22&hl=en&sa=X&ei=oJa6T5L2O6SriQKGloCUBw&ved=0CDQQ6AEwAA#v=onepage&q=%22Fourier%20introduced%20the%22%20%22%20-function%20much%20earlier%22&f=false |page=200 }}</ref>

:<math>\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty dp\ \cos (px-p\alpha) \ . </math>

Later, [[Augustin Cauchy]] expressed the theorem using exponentials:<ref name= Myint-U>{{cite book
|url=http://books.google.com/books?id=Zbz5_UvERIIC&pg=PA4&dq=%22It+was+the+work+of+Augustin+Cauchy%22&hl=en&sa=X&ei=RnW6T52LNovYiQLa9-mABw&ved=0CDgQ6AEwAA#v=onepage&q=%22It%20was%20the%20work%20of%20Augustin%20Cauchy%22&f=false |author=Tyn Myint-U., Lokenath Debnath |title=Linear Partial Differential Equations for Scientists And Engineers |isbn=0817643931 |edition=4th |year=2007 |page=4 |publisher=Springer}}</ref><ref name=Debnath>{{cite book |url=http://books.google.com/books?id=WbZcqdvCEfwC&pg=PA2&dq=%22It+was+the+work+of+Cauchy+that+contained%22&hl=en&sa=X&ei=Jym9T8L-NK6OigK-m_GYDg&ved=0CDQQ6AEwAA#v=onepage&q=%22It%20was%20the%20work%20of%20Cauchy%20that%20contained%22&f=false |title=Integral Transforms And Their Applications |author=Lokenath Debnath, Dambaru Bhatta |isbn=1584885750 |year=2007 |edition=2nd |publisher=CRC Press |page=2}}</ref>

:<math>f(x)=\frac{1}{2\pi} \int_{-\infty} ^ \infty \ e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\ d \alpha \right) \ dp. </math>

Cauchy pointed out that in some circumstances the ''order'' of integration in this result was significant.<ref name=Grattan-Guinness>{{cite book |title=Convolutions in French Mathematics, 1800–1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics, Volume 2 |page=653 |url= http://books.google.com/books?id=_GgioErrbW8C&pg=PA653&dq=%22Further,+in+a+double+integral%22&hl=en&sa=X&ei=4gC9T7KVDvDRiALq-dTLDQ&ved=0CDgQ6AEwAA#v=onepage&q=%22Further%2C%20in%20a%20double%20integral%22&f=false |isbn=3764322381 |year=2009 |publisher=Birkhäuser |author=Ivor Grattan-Guinness}}</ref><ref name=Cauchy>

See, for example, [http://gallica.bnf.fr/ark:/12148/bpt6k90181x/f387 ''Des intégrales doubles qui se présentent sous une forme indéterminèe'']</ref>

As justified using the [[Distribution (mathematics)|theory of distributions]], the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as:

:<math>\begin{align}
f(x)&=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\ d \alpha \right) \ dp \\
&=\frac{1}{2\pi} \int_{-\infty}^\infty \left(\int_{-\infty}^\infty e^{ipx} e^{-ip\alpha } \ dp \right)f(\alpha)\ d \alpha =\int_{-\infty}^\infty \delta (x-\alpha) f(\alpha) \ d \alpha,
\end{align}</math>

where the δ-function is expressed as:

:<math>\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\ dp \ . </math>

A rigorous interpretation of the exponential form and the various limitations upon the function ''f'' necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:<ref name="Mitrović"> {{cite book |title=Fundamentals of Applied Functional Analysis: Distributions, Sobolev Spaces |author=Dragiša Mitrović, Darko Žubrinić |url=http://books.google.com/books?id=Od5BxTEN0VsC&pg=PA62&dq=%22greatest+drawback+of+the+classical+Fourier+transformation+is+a+rather+narrow+class+of+functions%22&hl=en&sa=X&ei=IKG6T_niFqWfiQLJoODdBg&ved=0CDQQ6AEwAA#v=onepage&q=%22greatest%20drawback%20of%20the%20classical%20Fourier%20transformation%20is%20a%20rather%20narrow%20class%20of%20functions%22&f=false |page=62 |isbn=0582246946 |year=1998 |publisher=CRC Press}}</ref>
:The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) in order to insure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.

Further developments included generalization of the Fourier integral, "beginning with [[Michel Plancherel|Plancherel's]] pathbreaking ''L''2-theory (1910), continuing with [[Norbert Wiener|Wiener's]] and [[Salomon Bochner|Bochner's]] works (around 1930) and culminating with the amalgamation into [[Laurent Schwartz|L. Schwartz's]] theory of [[Distribution (mathematics)|distributions]] (1945)...",<ref name=Kracht>{{cite book |title=Topics in Mathematical Analysis: A Volume Dedicated to the Memory of A.L. Cauchy |url=http://books.google.com/books?id=xIsPrSiDlZIC&pg=PA553&dq=%22To+this+theory%22+%22and+even+more%22++%22that+one+was+able+to+generalize%22&hl=en&sa=X&ei=RJ66T-y7JOLjiAKuoeSUBw&ved=0CDQQ6AEwAA#v=onepage&q=%22To%20this%20theory%22%20%22and%20even%20more%22%20%20%22that%20one%20was%20able%20to%20generalize%22&f=false |author=Manfred Kracht, Erwin Kreyszig |page=553 |isbn=9971506661 |editor=Themistocles M. Rassias, ed |year=1989 |publisher=World Scientific |chapter=On singular integral operators and generalizations}}</ref> and leading to the formal development of the Dirac delta function.

An [[infinitesimal]] formula for an infinitely tall, unit impulse delta function (infinitesimal version of [[Cauchy distribution]]) explicitly appears in an 1827 text of [[Augustin Louis Cauchy]].<ref>{{harvnb|Laugwitz|1989|p=230}}</ref> [[Siméon Denis Poisson]] considered the issue in connection with the study of wave propagation as did [[Gustav Kirchhoff]] somewhat later. Kirchhoff and [[Hermann von Helmholtz]] also introduced the unit impulse as a limit of [[Gaussian distribution|Gaussians]], which also corresponded to [[Lord Kelvin]]'s notion of a point heat source. At the end of the 19th century, [[Oliver Heaviside]] used formal [[Fourier series]] to manipulate the unit impulse.<ref>A more complete historical account can be found in {{harvnb|van der Pol|Bremmer|1987|loc=§V.4}}.</ref> The Dirac delta function as such was introduced as a "convenient notation" by [[Paul Dirac]] in his influential 1930 book ''Principles of Quantum Mechanics''.<ref name="Dirac 1958 loc=§15">{{harvnb|Dirac|1958|loc=§15}}</ref> He called it the "delta function" since he used it as a continuous analogue of the discrete [[Kronecker delta]].

==Definitions==
The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

: <math>\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>

and which is also constrained to satisfy the identity

:<math>\int_{-\infty}^\infty \delta(x) \, dx = 1.</math><ref>{{harvnb|Gel'fand|Shilov|1968|loc=Volume I, §1.1, p. 1}}</ref>

This is merely a [[heuristic]] characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.<ref name="Dirac 1958 loc=§15"/> The Dirac delta function can be rigorously defined either as a [[distribution (mathematics)|distribution]] or as a [[measure (mathematics)|measure]].

===As a measure===
One way to rigorously define the delta function is as a [[Measure (mathematics)|measure]], which accepts as an argument a subset ''A'' of the real line '''R''', and returns δ(''A'') = 1 if 0 ∈ ''A'', and δ(''A'') = 0 otherwise.<ref name="Rudin 1966 loc=§1.20">{{harvnb|Rudin|1966|loc=§1.20}}</ref> If the delta function is conceptualized as modeling an idealized point mass at 0, then δ(''A'') represents the mass contained in the set ''A''. One may then define the integral against δ as the integral of a function against this mass distribution. Formally, the [[Lebesgue integral]] provides the necessary analytic device. The Lebesgue integral with respect to the measure δ satisfies

: <math>\int_{-\infty}^\infty f(x) \, \delta\{dx\} = f(0)</math>

for all continuous compactly supported functions ''f''. The measure δ is not [[absolutely continuous]] with respect to the [[Lebesgue measure]] — in fact, it is a [[singular measure]]. Consequently, the delta measure has no [[Radon–Nikodym derivative]] — no true function for which the property

:<math>\int_{-\infty}^\infty f(x)\delta(x)\, dx = f(0)</math>

holds.<ref>{{harvnb|Hewitt|Stromberg|1963|loc=§19.61}}</ref> As a result, the latter notation is a convenient [[abuse of notation]], and not a standard ([[Riemann integral|Riemann]] or [[Lebesgue integral|Lebesgue]]) integral.

As a [[probability measure]] on '''R''', the delta measure is characterized by its [[cumulative distribution function]], which is the [[unit step function]]<ref>{{harvnb|Driggers|2003|p=2321}}. See also {{harvnb|Bracewell|1986|loc=Chapter 5}} for a different interpretation. Other conventions for the assigning the value of the Heaviside function at zero exist, and some of these are not consistent with what follows.</ref>

:<math>H(x) =
\begin{cases}
1 & \text{if } x\ge 0\\
0 & \text{if } x < 0.
\end{cases}</math>

This means that ''H''(''x'') is the integral of the cumulative [[indicator function]] '''1'''(−∞, ''x''] with respect to the measure δ; to wit,

:<math>H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta\{dt\} = \delta(-\infty,x].</math>

Thus in particular the integral of the delta function against a continuous function can be properly understood as a [[Stieltjes integral]]:<ref>{{harvnb|Hewitt|Stromberg|1965|loc=§9.19}}</ref>

:<math>\int_{-\infty}^\infty f(x)\delta\{dx\} = \int_{-\infty}^\infty f(x) \, dH(x).</math>

All higher [[moment (mathematics)|moments]] of δ are zero. In particular, [[characteristic function (probability theory)|characteristic function]] and [[moment generating function]] are both equal to one.

===As a distribution===
In the theory of [[distribution (mathematics)|distributions]] a generalized function is thought of not as a function itself, but only in relation to how it affects other functions when it is "integrated" against them. In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function against a sufficiently "good" test function is. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.

A typical space of test functions consists of all [[smooth function]]s on '''R''' with [[compact support]]. As a distribution, the Dirac delta is a [[linear functional]] on the space of test functions and is defined by<ref>{{harvnb|Strichartz|1994|loc=§2.2}}</ref>

{{NumBlk|:| <math>\delta[\varphi] = \varphi(0)\,</math>|{{EquationRef|1}}}}

for every test function φ.

For δ to be properly a distribution, it must be "continuous" in a suitable sense. In general, for a linear functional ''S'' on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer ''N'' there is an integer ''M''''N'' and a constant ''C''''N'' such that for every test function φ, one has the inequality<ref>{{harvnb|Hörmander|1983|loc=Theorem 2.1.5}}</ref>

:<math>|S[\phi]| \le C_N \sum_{k=0}^{M_N}\sup_{x\in [-N,N]}|\phi^{(k)}(x)|.</math>

With the δ distribution, one has such an inequality (with ''C''''N'' = 1) with ''M''''N'' = 0 for all ''N''. Thus δ is a distribution of order zero. It is, furthermore, a distribution with compact support (the [[support (mathematics)|support]] being {0}).

The delta distribution can also be defined in a number of equivalent ways. For instance, it is the [[distributional derivative]] of the [[Heaviside step function]]. This means that, for every test function φ, one has

:<math>\delta[\phi] = -\int_{-\infty}^\infty \phi'(x)H(x)\, dx.</math>

Intuitively, if [[integration by parts]] were permitted, then the latter integral should simplify to

:<math>\int_{-\infty}^\infty \phi(x)H'(x)\, dx = \int_{-\infty}^\infty \phi(x)\delta(x)\, dx,</math>

and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case one does have

:<math>-\int_{-\infty}^\infty \phi'(x)H(x)\, dx = \int_{-\infty}^\infty \phi(x)\,dH(x).</math>

In the context of measure theory, the Dirac measure gives rise to a distribution by integration. Conversely, equation ({{EquationNote|1}}) defines a [[Daniell integral]] on the space of all compactly supported continuous functions φ which, by the [[Riesz representation theorem]], can be represented as the Lebesgue integral of φ with respect to some [[Radon measure]].

===Generalizations===
The delta function can be defined in ''n''-dimensional [[Euclidean space]] '''R'''''n'' as the measure such that

:<math>\int_{\mathbf{R}^n} f(\mathbf{x})\delta\{d\mathbf{x}\} = f(\mathbf{0})</math>

for every compactly supported continuous function ''f''. As a measure, the ''n''-dimensional delta function is the [[product measure]] of the 1-dimensional delta functions in each variable separately. Thus, formally, with '''x''' = (''x''1,''x''2,...,''x''''n''), one has<ref name="Bracewell 1986 loc=Chapter 5"/>

{{NumBlk|:|<math>\delta(\mathbf{x}) = \delta(x_1)\delta(x_2)\dots\delta(x_n).</math>|{{EquationRef|2}}}}

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.<ref>{{harvnb|Hörmander|1983|loc=§3.1}}</ref> However, despite widespread use in engineering contexts, ({{EquationNote|2}}) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.<ref>{{harvnb|Strichartz|1994|loc=§2.3}}; {{harvnb|Hörmander|1983|loc=§8.2}}</ref>

The notion of a '''[[Dirac measure]]''' makes sense on any set whatsoever.<ref name="Rudin 1966 loc=§1.20"/> Thus if ''X'' is a set, ''x''0 ∈ ''X'' is a marked point, and Σ is any [[sigma algebra]] of subsets of ''X'', then the measure defined on sets ''A'' ∈ Σ by

:<math>\delta_{x_0}(A)=\begin{cases}
1 &\rm{if\ }x_0\in A\\
0 &\rm{if\ }x_0\notin A
\end{cases}</math>

is the delta measure or unit mass concentrated at ''x''0.

Another common generalization of the delta function is to a [[differentiable manifold]] where most of its properties as a distribution can also be exploited because of the [[differentiable structure]]. The delta function on a manifold ''M'' centered at the point ''x''0 ∈ ''M'' is defined as the following distribution:

{{NumBlk|:|<math>\delta_{x_0}[\phi] = \phi(x_0)</math>|{{EquationRef|3}}}}

for all compactly supported smooth real-valued functions φ on ''M''.<ref>{{harvnb|Dieudonné|1972|loc=§17.3.3}}</ref> A common special case of this construction is when ''M'' is an [[open set]] in the Euclidean space '''R'''''n''.

On a [[locally compact Hausdorff space]] ''X'', the Dirac delta measure concentrated at a point ''x'' is the [[Radon measure]] associated with the Daniell integral ({{EquationNote|3}}) on compactly supported continuous functions φ. At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping <math>x_0\mapsto \delta_{x_0}</math> is a continuous embedding of ''X'' into the space of finite Radon measures on ''X'', equipped with its [[vague topology]]. Moreover, the [[convex hull]] of the image of ''X'' under this embedding is [[dense set|dense]] in the space of probability measures on ''X''.<ref>{{harvnb|Federer|1969|loc=§2.5.19}}</ref>

==Properties==
===Scaling and symmetry===
The delta function satisfies the following scaling property for a non-zero scalar α:<ref>{{harvnb|Strichartz|1994|loc=Problem 2.6.2}}</ref>

:<math>\int_{-\infty}^\infty \delta(\alpha x)\,dx
=\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|}
=\frac{1}{|\alpha|}</math>

and so

{{NumBlk|:|<math>\delta(\alpha x) = \frac{\delta(x)}{|\alpha|}.</math>|{{EquationRef|4}}}}

In particular, the delta function is an [[even function|even]] distribution, in the sense that

:<math>\delta(-x) = \delta(x)</math>

which is [[homogeneous function|homogeneous]] of degree −1.

===Algebraic properties===
The [[distribution (mathematics)|distributional product]] of δ with ''x'' is equal to zero:

:<math>x\delta(x) = 0.</math>

Conversely, if ''xf''(''x'') = ''xg''(''x''), where ''f'' and ''g'' are distributions, then

:<math>f(x) = g(x) +c \delta(x)</math>

for some constant ''c''.<ref>{{harvnb|Vladimirov|1971|loc=Chapter 2, Example 3(d)}}</ref>

===Translation===
The integral of the time-delayed Dirac delta is given by''':'''

:<math>\int_{-\infty}^\infty f(t) \delta(t-T)\,dt = f(T).</math>

This is sometimes referred to as the ''sifting property''<ref>{{MathWorld|urlname=SiftingProperty|title=Sifting Property}}</ref> or the ''sampling property''. The delta function is said to "sift out" the value at ''t'' = ''T''.

It follows that the effect of [[Convolution|convolving]] a function ''f''(''t'') with the time-delayed Dirac delta is to time-delay ''f''(''t'') by the same amount''':'''

:{|
|<math>(f(t) * \delta(t-T))\,</math>
|<math> \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty f(\tau) \delta(t-T-\tau) \, d\tau</math>
|-
|
|<math>= \int\limits_{-\infty}^\infty f(\tau) \delta(\tau-(t-T)) \, d\tau</math>       (using  ({{EquationNote|4}}): <math>\delta(-x)=\delta(x)</math>)
|-
|
|<math>= f(t-T).\,</math>
|}

This holds under the precise condition that ''f'' be a [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]] (see the discussion of the Fourier transform [[#Fourier transform|below]]). As a special case, for instance, we have the identity (understood in the distribution sense)

:<math>\int_{-\infty}^\infty \delta (\xi-x) \delta(x-\eta) \, dx = \delta(\xi-\eta).</math>

===Composition with a function===
More generally, the delta distribution may be [[distribution (mathematics)#Composition with a smooth function|composed]] with a smooth function ''g''(''x'') in such a way that the familiar change of variables formula holds, that

:<math>\int_{\mathbf{R}} \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) |g'(x)|\,dx = \int_{g(\mathbf{R})} \delta(u)f(u)\, du</math>

provided that ''g'' is a [[continuously differentiable]] function with ''g''′ nowhere zero.<ref name="ReferenceA">{{harvnb|Gel'fand|Shilov|1966–1968|loc=Vol. 1, §II.2.5}}</ref> That is, there is a unique way to assign meaning to the distribution <math>\delta\circ g</math> so that this identity holds for all compactly supported test functions ''f''. This distribution satisfies δ(''g''(''x'')) = 0 if ''g'' is nowhere zero, and otherwise if ''g'' has a real [[root of a function|root]] at ''x''0, then

: <math>\delta(g(x)) = \frac{\delta(x-x_0)}{|g'(x_0)|}.</math>

It is natural therefore to ''define'' the composition δ(''g''(''x'')) for continuously differentiable functions ''g'' by

:<math>\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}</math>

where the sum extends over all roots of ''g''(''x''), which are assumed to be simple.<ref name="ReferenceA"/> Thus, for example

:<math>\delta\left(x^2-\alpha^2\right) = \frac{1}{2|\alpha|}\Big[\delta\left(x+\alpha\right)+\delta\left(x-\alpha\right)\Big].</math>

In the integral form the generalized scaling property may be written as

: <math> \int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}. </math>

===Properties in ''n'' dimensions===
The delta distribution in an ''n''-dimensional space satisfies the following scaling property instead:

:<math>\delta(\alpha\mathbf{x}) = |\alpha|^{-n}\delta(\mathbf{x})</math>

so that δ is a [[homogeneous function|homogeneous]] distribution of degree −''n''. Under any [[reflection (mathematics)|reflection]] or [[rotation (mathematics)|rotation]] ρ, the delta function is invariant:

:<math>\delta(\rho \mathbf{x}) = \delta(\mathbf{x}).</math>

As in the one-variable case, it is possible to define the composition of δ with a [[Lipschitz function|bi-Lipschitz function]]<ref>Further refinement is possible, namely to [[submersion (mathematics)|submersions]], although these require a more involved change of variables formula.</ref> ''g'': '''R'''''n'' → '''R'''''n'' uniquely so that the identity

:<math>\int_{\mathbf{R}^n} \delta(g(\mathbf{x}))\, f(g(\mathbf{x}))\, |\det g'(\mathbf{x})|\, d\mathbf{x} = \int_{g(\mathbf{R}^n)} \delta(\mathbf{u}) f(\mathbf{u})\,d\mathbf{u}</math>

for all compactly supported functions ''f''.

Using the [[coarea formula]] from [[geometric measure theory]], one can also define the composition of the delta function with a [[submersion (mathematics)|submersion]] from one Euclidean space to another one of different dimension; the result is a type of [[current (mathematics)|current]]. In the special case of a continuously differentiable function ''g'': '''R'''''n'' → '''R''' such that the [[gradient]] of ''g'' is nowhere zero, the following identity holds<ref>{{harvnb|Hörmander|1983|loc=§6.1}}</ref>

: <math>\int_{\mathbf{R}^n} f(\mathbf{x}) \, \delta(g(\mathbf{x})) \, d\mathbf{x} = \int_{g^{-1}(0)}\frac{f(\mathbf{x})}{|\mathbf{\nabla}g|}\,d\sigma(\mathbf{x}) </math>

where the integral on the right is over ''g''−1(0), the ''n'' − 1 dimensional surface defined by ''g''('''x''') = 0 with respect to the [[Minkowski content]] measure. This is known as a [[simple layer]] integral.

More generally, if ''S'' is a smooth hypersurface of '''R'''''n'', then we can associated to ''S'' the distribution that integrates any compactly supported smooth function ''g'' over ''S'':
:<math>\delta_S[g] = \int_S g(\mathbf{s})\,d\sigma(\mathbf{s})</math>

where σ is the hypersurface measure associated to ''S''. This generalization is associated with the [[potential theory]] of [[simple layer potential]]s on ''S''. If ''D'' is a [[domain (mathematical analysis)|domain]] in '''R'''''n'' with smooth boundary ''S'', then δ''S'' is equal to the [[normal derivative]] of the [[indicator function]] of ''D'' in the distribution sense:

:<math>-\int_{\mathbf{R}^n}g(\mathbf{x})\,\frac{\partial 1_D(\mathbf{x})}{\partial n}\;d\mathbf{x}=\int_S\,g(\mathbf{s})\;d\sigma(\mathbf{s}),</math>

where ''n'' is the outward normal.<ref>{{harvnb|Lange|2012|loc=pp.29–30}}</ref><ref>{{harvnb|Gelfand|Shilov|p=212}}</ref> For a proof, see e.g. the article on the [[Laplacian of the indicator#Surface Dirac delta function|normal derivative of the indicator function]].

==Fourier transform==
The delta function is a [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]], and therefore it has a well-defined [[Fourier transform]]. Formally, one finds<ref>In some conventions for the Fourier transform.</ref>

:<math>\hat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi}\delta(x)\,dx = 1.</math>

Properly speaking, the Fourier transform of a distribution is defined by imposing [[self-adjoint]]ness of the Fourier transform under the duality pairing <math>\langle\cdot,\cdot\rangle</math> of tempered distributions with [[Schwartz functions]]. Thus <math>\hat{\delta}</math> is defined as the unique tempered distribution satisfying

:<math>\langle\hat{\delta},\phi\rangle = \langle\delta,\hat{\phi}\rangle</math>

for all Schwartz functions φ. And indeed it follows from this that <math>\hat{\delta}=1.</math>

As a result of this identity, the [[convolution]] of the delta function with any other tempered distribution ''S'' is simply ''S'':

:<math>S*\delta = S.\,</math>

That is to say that δ is an [[identity element]] for the convolution on tempered distributions, and in fact the space of compactly supported distributions under convolution is an [[associative algebra]] with identity the delta function. This property is fundamental in [[signal processing]], as convolution with a tempered distribution is a [[linear time-invariant system]], and applying the linear time-invariant system measures its [[impulse response]]. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for δ, and once it is known, it characterizes the system completely. See [[LTI system theory#Impulse response and convolution|''LTI system theory:Impulse response and convolution'']].

The inverse Fourier transform of the tempered distribution ''f''(ξ) = 1 is the delta function. Formally, this is expressed

:<math>\int_{-\infty}^\infty 1 \cdot e^{2\pi i x\xi}\,d\xi = \delta(x)</math>

and more rigorously, it follows since

:<math>\langle 1, f^\vee\rangle = f(0) = \langle\delta,f\rangle</math>

for all Schwartz functions ''f''.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on '''R'''. Formally, one has

:<math>\int_{-\infty}^\infty e^{i 2\pi \xi_1 t} \left[e^{i 2\pi \xi_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (\xi_2 - \xi_1) t} \,dt = \delta(\xi_2 - \xi_1).</math>

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution

:<math>f(t) = e^{i2\pi\xi_1 t}</math>

is

:<math>\hat{f}(\xi_2) = \delta(\xi_1-\xi_2)</math>

which again follows by imposing self-adjointness of the Fourier transform.

By [[analytic continuation]] of the Fourier transform, the [[Laplace transform]] of the delta function is found to be<ref>{{harvnb|Bracewell|1986}}</ref>

: <math> \int_{0}^{\infty}\delta (t-a)e^{-st} \, dt=e^{-sa}.</math>

==Distributional derivatives==
The distributional derivative of the Dirac delta distribution is the distribution δ′ defined on compactly supported smooth test functions φ by<ref>{{harvnb|Gel'fand|Shilov|1966|p=26}}</ref>

:<math>\delta'[\varphi] = -\delta[\varphi']=-\varphi'(0).</math>

The first equality here is a kind of integration by parts, for if δ were a true function then

:<math>\int_{-\infty}^\infty \delta'(x)\varphi(x)\,dx = -\int_{-\infty}^\infty \delta(x)\varphi'(x)\,dx.</math>

The ''k''-th derivative of δ is defined similarly as the distribution given on test functions by

:<math>\delta^{(k)}[\varphi] = (-1)^k \varphi^{(k)}(0).</math>

In particular δ is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:<ref>{{harvnb|Gel'fand|Shilov|1966|loc=§2.1}}</ref>

:<math>\delta'(x) = \lim_{h\to 0} \frac{\delta(x+h)-\delta(x)}{h}.</math>

More properly, one has

:<math>\delta' = \lim_{h\to 0} \frac{1}{h}(\tau_h\delta - \delta)</math>

where τ''h'' is the translation operator, defined on functions by τ''h''φ(x) = φ(x+h), and on a distribution ''S'' by

:<math>(\tau_h S)[\varphi] = S[\tau_{-h}\varphi].</math>

In the theory of [[electromagnetism]], the first derivative of the delta function represents a point magnetic [[dipole]] situated at the origin. Accordingly, it is referred to as a dipole or the [[unit doublet|doublet function]].<ref>{{MathWorld|title=Doublet Function|urlname=DoubletFunction}}</ref>

The derivative of the delta function satisfies a number of basic properties, including:

*<math>\frac{d}{dx}\delta(-x) = \frac{d}{dx}\delta(x)</math>
*<math>\delta'(-x) = -\delta'(x)</math>
*<math>x\delta'(x) = -\delta(x).</math><ref>The property follows by applying a test function and integration by parts.</ref>

Furthermore, the convolution of δ' with a compactly supported smooth function ''f'' is

:<math>\delta'*f = \delta*f' = f',</math>

which follows from the properties of the distributional derivative of a convolution.

===Higher dimensions===
More generally, on an [[open set]] ''U'' in the ''n''-dimensional [[Euclidean space]] '''R'''''n'', the Dirac delta distribution centered at a point ''a'' ∈ ''U'' is defined by<ref name="Hörmander 1983 56">{{harvnb|Hörmander|1983|p=56}}</ref>

:<math>\delta_a[\phi]=\phi(a)</math>

for all φ ∈ ''S''(''U''), the space of all smooth compactly supported functions on ''U''. If ''α'' = (α1, ..., α''n'') is any [[multi-index]] and ∂α denotes the associated mixed [[partial derivative]] operator, then the αth derivative ∂αδ''a'' of δ''a'' is given by<ref name="Hörmander 1983 56"/>

:<math>\left\langle \partial^{\alpha} \delta_{a}, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = \left. (-1)^{| \alpha |} \partial^{\alpha} \varphi (x) \right|_{x = a} \mbox{ for all } \varphi \in S(U).</math>

That is, the αth derivative of δ''a'' is the distribution whose value on any test function φ is the αth derivative of φ at ''a'' (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as [[double layer potential|double layers]] along the coordinate planes. More generally, the [[normal derivative]] of a simple layer supported on a surface is a double layer supported on that surface, and represents a laminar magnetic monopole. Higher derivatives of the delta function are known in physics as [[multipole]]s.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If ''S'' is any distribution on ''U'' supported on the set {''a''} consisting of a single point, then there is an integer ''m'' and coefficients ''c''α such that<ref>{{harvnb|Hörmander|1983|p=56}}; {{harvnb|Rudin|1991|loc=Theorem 6.25}}</ref>

:<math>S = \sum_{|\alpha|\le m} c_\alpha \partial^\alpha\delta_a.</math>

==Representations of the delta function==
The delta function can be viewed as the limit of a sequence of functions

:<math>\delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x), \,</math>

where ηε(''x'') is sometimes called a '''nascent delta function'''{{anchor|nascent delta function}}. This limit is meant in a weak sense: either that

{{NumBlk|:|<math> \lim_{\varepsilon\to 0^+} \int_{-\infty}^{\infty}\eta_\varepsilon(x)f(x) \, dx = f(0) \ </math>|{{EquationRef|5}}}}

for all [[continuous function|continuous]] functions ''f'' having [[compact support]], or that this limit holds for all [[smooth function|smooth]] functions ''f'' with compact support. The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the [[vague topology]] of measures, and the latter is convergence in the sense of [[distribution (mathematics)|distributions]].

===Approximations to the identity===
Typically a nascent delta function ηε can be constructed in the following manner. Let η be an absolutely integrable function on '''R''' of total integral 1, and define

:<math>\eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). </math>

In ''n'' dimensions, one uses instead the scaling

:<math>\eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). </math>

Then a simple change of variables shows that ηε also has integral 1.<ref>{{harvnb|Stein|Weiss|loc=Theorem 1.18}}</ref> One shows easily that ({{EquationNote|5}}) holds for all continuous compactly supported functions ''f'', and so ηε converges weakly to δ in the sense of measures.

The ηε constructed in this way are known as an '''approximation to the identity'''.<ref>{{harvnb|Rudin|1991|loc=§II.6.31}}</ref> This terminology is because the space ''L''1('''R''') of absolutely integrable functions is closed under the operation of [[convolution]] of functions: ''f''∗''g'' ∈ ''L''1('''R''') whenever ''f'' and ''g'' are in ''L''1('''R'''). However, there is no identity in ''L''1('''R''') for the convolution product: no element ''h'' such that ''f''∗''h'' = ''f'' for all ''f''. Nevertheless, the sequence ηε does approximate such an identity in the sense that

:<math>f*\eta_\varepsilon \to f\quad\rm{as\ }\varepsilon\to 0.</math>

This limit holds in the sense of [[mean convergence]] (convergence in ''L''1). Further conditions on the ηε, for instance that it be a mollifier associated to a compactly supported function,<ref>More generally, one only needs η = η1 to have an integrable radially symmetric decreasing rearrangement.</ref> are needed to ensure pointwise convergence [[almost everywhere]].

If the initial η = η1 is itself smooth and compactly supported then the sequence is called a [[mollifier]]. The standard mollifier is obtained by choosing η to be a suitably normalized [[bump function]], for instance

:<math>\eta(x) = \begin{cases} e^{-\frac{1}{1-|x|^2}}& \text{ if } |x| < 1\\
0& \text{ if } |x|\geq 1.
\end{cases}</math>

In some situations such as [[numerical analysis]], a [[piecewise linear function|piecewise linear]] approximation to the identity is desirable. This can be obtained by taking η1 to be a [[hat function]]. With this choice of η1, one has

:<math> \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-|\frac{x}{\varepsilon}|,0 \right) </math>

which are all continuous and compactly supported, although not smooth and so not a mollifier.

===Probabilistic considerations===
In the context of [[probability theory]], it is natural to impose the additional condition that the initial η1 in an approximation to the identity should be positive, as such a function then represents a [[probability distribution]]. Convolution with a probability distribution is sometimes favorable because it does not result in [[overshoot (signal)|overshoot]] or undershoot, as the output is a [[convex combination]] of the input values, and thus falls between the maximum and minimum of the input function. Taking η1 to be any probability distribution at all, and letting ηε(''x'') = η1(''x''/ε)/ε as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments. For instance, if η1 is the [[uniform distribution (continuous)|uniform distribution]] on [−1/2, 1/2], also known as the [[rectangular function]], then:<ref>{{harvnb|Saichev|Woyczyński|1997|loc=§1.1 The "delta function" as viewed by a physicist and an engineer, p. 3}}</ref>

:<math>\eta_\varepsilon(x) = \frac{1}{\varepsilon}\ \textrm{rect}\left(\frac{x}{\varepsilon}\right)=
\begin{cases}
\frac{1}{\varepsilon},&-\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2}\\
0,&\text{otherwise}.
\end{cases}</math>

Another example is with the [[Wigner semicircle distribution]]

:<math>\eta_\varepsilon(x)= \begin{cases}
\frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon < x < \varepsilon \\
0, & \text{otherwise}
\end{cases}</math>

This is continuous and compactly supported, but not a mollifier because it is not smooth.

===Semigroups===
Nascent delta functions often arise as convolution [[semigroup]]s. This amounts to the further constraint that the convolution of ηε with ηδ must satisfy

:<math>\eta_\varepsilon * \eta_\delta = \eta_{\varepsilon+\delta}</math>

for all ε, δ > 0. Convolution semigroups in ''L''1 that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction.

In practice, semigroups approximating the delta function arise as [[fundamental solution]]s or [[Green's function]]s to physically motivated [[elliptic partial differential equation|elliptic]] or [[parabolic partial differential equation|parabolic]] [[partial differential equations]]. In the context of [[applied mathematics]], semigroups arise as the output of a [[linear time-invariant system]]. Abstractly, if ''A'' is a linear operator acting on functions of ''x'', then a convolution semigroup arises by solving the [[initial value problem]]

:<math>\begin{cases}
\frac{\partial}{\partial t}\eta(t,x) = A\eta(t,x), \quad t>0 \\
\displaystyle\lim_{t\to 0^+} \eta(t,x) = \delta(x)
\end{cases}</math>

in which the limit is as usual understood in the weak sense. Setting ηε(''x'') = η(ε, ''x'') gives the associated nascent delta function.

Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.

;The heat kernel
The [[heat kernel]], defined by

:<math>\eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}}</math>

represents the temperature in an infinite wire at time ''t'' > 0, if a unit of heat energy is stored at the origin of the wire at time ''t'' = 0. This semigroup evolves according to the one-dimensional [[heat equation]]:

:<math>\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}.</math>

In [[probability theory]], ηε(''x'') is a [[normal distribution]] of [[variance]] ε and mean 0. It represents the [[probability density function|probability density]] at time ''t'' = ε of the position of a particle starting at the origin following a standard [[Brownian motion]]. In this context, the semigroup condition is then an expression of the [[Markov property]] of Brownian motion.

In higher dimensional Euclidean space '''R'''''n'', the heat kernel is

:<math>\eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}},</math>

and has the same physical interpretation, ''[[mutatis mutandis]]''. It also represents a nascent delta function in the sense that ηε → δ in the distribution sense as ε → 0.

;The Poisson kernel
The [[Poisson kernel]]

:<math>\eta_\varepsilon(x) = \frac{1}{\pi} \frac{\varepsilon}{\varepsilon^2 + x^2}=\int_{-\infty}^{\infty}\mathrm{e}^{2\pi\mathrm{i} \xi x-|\varepsilon \xi|}\;d\xi</math>

is the fundamental solution of the [[Laplace equation]] in the upper half-plane.<ref>{{harvnb|Stein|Weiss|1971|loc=§I.1}}</ref> It represents the [[electrostatic potential]] in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the [[Cauchy distribution]]. This semigroup evolves according to the equation

:<math>\frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x)</math>

where the operator is rigorously defined as the [[Fourier multiplier]]

:<math>\mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).</math>

===Oscillatory integrals===
In areas of physics such as [[wave propagation]] and [[wave|wave mechanics]], the equations involved are [[hyperbolic partial differential equations|hyperbolic]] and so may have more singular solutions. As a result, the nascent delta functions that arise as fundamental solutions of the associated [[Cauchy problem]]s are generally [[oscillatory integral]]s. An example, which comes from a solution of the [[Euler–Tricomi equation]] of [[transonic]] [[gas dynamics]],<ref>{{harvnb|Vallée|Soares|2004|loc=§7.2}}</ref> is the rescaled [[Airy function]]

:<math>\varepsilon^{-\frac{1}{3}}\operatorname{Ai}\left (x\varepsilon^{-\frac{1}{3}} \right). </math>

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense, it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the [[Dirichlet kernel]] below), rather than in the sense of measures.

Another example is the Cauchy problem for the [[wave equation]] in '''R'''1+1:<ref>{{harvnb|Hörmander|1983|loc=§7.8}}</ref>

:<math> \begin{align}
c^{-2}\frac{\partial^2u}{\partial t^2} - \Delta u &= 0\\
u=0,\quad \frac{\partial u}{\partial t} = \delta &\qquad \text{for }t=0.
\end{align} </math>

The solution ''u'' represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin.

Other approximations to the identity of this kind include the [[sinc function]] (used widely in electronics and telecommunications)

:<math>\eta_\varepsilon(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=\frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}} \cos(kx)\;dk </math>

and the [[Bessel function]]

:<math> \eta_\varepsilon(x) = \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right). </math>

===Plane wave decomposition===
One approach to the study of a linear partial differential equation

:<math>L[u]=f,\,</math>

where ''L'' is a [[differential operator]] on '''R'''''n'', is to seek first a fundamental solution, which is a solution of the equation

:<math>L[u]=\delta.\,</math>

When ''L'' is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form

:<math>L[u]=h\,</math>

where ''h'' is a [[plane wave]] function, meaning that it has the form

:<math>h = h(x\cdot\xi)</math>

for some vector ξ. Such an equation can be resolved (if the coefficients of ''L'' are [[analytic function]]s) by the [[Cauchy–Kovalevskaya theorem]] or (if the coefficients of ''L'' are constant) by quadrature. So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations.

Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by [[Johann Radon]], and then developed in this form by [[Fritz John]] ([[#CITEREFJohn1955|1955]]).<ref>See also {{harvnb|Courant|Hilbert|1962|loc=§14}}.</ref> Choose ''k'' so that ''n'' + ''k'' is an even integer, and for a real number ''s'', put

:<math>g(s) = \operatorname{Re}\left[\frac{-s^k\log(-is)}{k!(2\pi i)^n}\right]
=\begin{cases}
\frac{|s|^k}{4k!(2\pi i)^{n-1}}&n \text{ odd}\\
&\\
-\frac{|s|^k\log|s|}{k!(2\pi i)^{n}}&n \text{ even.}
\end{cases}</math>

Then δ is obtained by applying a power of the [[Laplacian]] to the integral with respect to the unit [[sphere measure]] dω of ''g''(''x'' · ξ) for ξ in the [[unit sphere]] ''S''''n''−1:

:<math>\delta(x) = \Delta_x^{\frac{n+k}{2}} \int_{S^{n-1}} g(x\cdot\xi)\,d\omega_\xi.</math>

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ,

:<math>\varphi(x) = \int_{\mathbf{R}^n}\varphi(y)\,dy\,\Delta_x^{\frac{n+k}{2}} \int_{S^{n-1}} g((x-y)\cdot\xi)\,d\omega_\xi.</math>

The result follows from the formula for the [[Newtonian potential]] (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the [[Radon transform]], because it recovers the value of φ(''x'') from its integrals over hyperplanes. For instance, if ''n'' is odd and ''k'' = 1, then the integral on the right hand side is

:<math>c_n \Delta^{\frac{n+1}{2}}_x\int\int_{S^{n-1}} \varphi(y)|(y-x)\cdot\xi|\,d\omega_\xi\,dy = c_n\Delta^{\frac{n+1}{2}}_x\int_{S^{n-1}} \, d\omega_\xi \int_{-\infty}^\infty |p|R\varphi(\xi,p+x\cdot\xi)\,dp</math>

where ''R''φ(ξ, ''p'') is the Radon transform of φ:

:<math>R\varphi(\xi,p) = \int_{x\cdot\xi=p} f(x)\,d^{n-1}x.</math>

An alternative equivalent expression of the plane wave decomposition, from {{harvtxt|Gel'fand|Shilov|1966–1968|loc=I, §3.10}}, is

:<math>\delta(x) = \frac{(n-1)!}{(2\pi i)^n}\int_{S^{n-1}}(x\cdot\xi)^{-n}\,d\omega_\xi</math>

for ''n'' even, and

:<math>\delta(x) = \frac{1}{2(2\pi i)^{n-1}}\int_{S^{n-1}}\delta^{(n-1)}(x\cdot\xi)\,d\omega_\xi</math>

for ''n'' odd.

===Fourier kernels===
{{See also|Convergence of Fourier series}}
In the study of [[Fourier series]], a major question consists of determining whether and in what sense the Fourier series associated with a [[periodic function]] converges to the function. The ''n''th partial sum of the Fourier series of a function ''f'' of period 2π is defined by convolution (on the interval [−π,π]) with the [[Dirichlet kernel]]:
:<math>D_N(x) = \sum_{n=-N}^N e^{inx} = \frac{\sin\left((N+\tfrac12)x\right)}{\sin(x/2)}.</math>
Thus,
:<math>s_N(f)(x) = D_N*f(x) = \sum_{n=-N}^N a_n e^{inx}</math>
where
:<math>a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy.</math>
A fundamental result of elementary Fourier series states that the Dirichlet kernel tends to the a multiple of the delta function as ''N'' → ∞. This is interpreted in the distribution sense, that
:<math>s_N(f)(0) = \int_{\mathbf{R}} D_N(x)f(x)\,dx \to 2\pi f(0)</math>
for every compactly supported ''smooth'' function ''f''. Thus, formally one has
:<math>\delta(x) = \frac1{2\pi} \sum_{n=-\infty}^\infty e^{inx}</math>
on the interval [−π,π].

In spite of this, the result does not hold for all compactly supported ''continuous'' functions: that is ''DN'' does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of [[summability methods]] in order to produce convergence. The method of [[Cesàro summation]] leads to the [[Fejér kernel]]<ref>{{harvnb|Lang|1997|p=312}}</ref>

:<math>F_N(x) = \sum_{n=0}^N D_n(x) = \frac{1}{N}\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)^2.</math>

The [[Fejér kernel]]s tend to the delta function in a stronger sense that<ref>In the terminology of {{harvtxt|Lang|1997}}, the Fejér kernel is a Dirac sequence, whereas the Dirichlet kernel is not.</ref>

:<math>\int_{\mathbf{R}} F_N(x)f(x)\,dx \to 2\pi f(0)</math>

for every compactly supported ''continuous'' function ''f''. The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point.

===Hilbert space theory===
The Dirac delta distribution is a [[densely defined]] [[unbounded operator|unbounded]] [[linear functional]] on the [[Hilbert space]] [[Lp space|L2]] of [[square integrable function]]s. Indeed, smooth compactly support functions are [[dense set|dense]] in ''L''2, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of ''L''2 and to give a stronger [[topology]] on which the delta function defines a [[bounded linear functional]].

;Sobolev spaces
The [[Sobolev embedding theorem]] for [[Sobolev space]]s on the real line '''R''' implies that any square-integrable function ''f'' such that

:<math>\|f\|_{H^1}^2 = \int_{-\infty}^\infty |\hat{f}(\xi)|^2 (1+|\xi|^2)\,d\xi < \infty</math>

is automatically continuous, and satisfies in particular

:<math>\delta[f]=|f(0)| < C \|f\|_{H^1}.</math>

Thus δ is a bounded linear functional on the Sobolev space ''H''1. Equivalently δ is an element of the [[continuous dual space]] ''H''−1 of ''H''1. More generally, in ''n'' dimensions, one has {{nowrap|δ ∈ ''H''−''s''('''R'''''n'')}} provided {{nowrap|''s'' > ''n'' / 2}}.

====Spaces of holomorphic functions====
In [[complex analysis]], the delta function enters via [[Cauchy's integral formula]] which asserts that if ''D'' is a domain in the [[complex plane]] with smooth boundary, then

:<math>f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D</math>

for all [[holomorphic function]]s ''f'' in ''D'' that are continuous on the closure of ''D''. As a result, the delta function δ''z'' is represented on this class of holomorphic functions by the Cauchy integral:

:<math>\delta_z[f] = f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z}.</math>

More generally, let ''H''2(∂''D'') be the [[Hardy space]] consisting of the closure in [[Lp space|''L''2(∂''D'')]] of all holomorphic functions in ''D'' continuous up to the boundary of ''D''. Then functions in ''H''2(∂''D'') uniquely extend to holomorphic functions in ''D'', and the Cauchy integral formula continues to hold. In particular for ''z'' ∈ ''D'', the delta function δ''z'' is a continuous linear functional on ''H''2(∂''D''). This is a special case of the situation in [[several complex variables]] in which, for smooth domains ''D'', the [[Szegő kernel]] plays the role of the Cauchy integral.

====Resolutions of the identity====
Given a complete [[orthonormal basis]] set of functions {φ''n''} in a separable Hilbert space, for example, the normalized [[eigenvector]]s of a [[Compact operator on Hilbert space#Spectral theorem|compact self-adjoint operator]], any vector ''f'' can be expressed as:
:<math>f = \sum_{n=1}^\infty \alpha_n \varphi_n. </math>
The coefficients {αn} are found as:
:<math>\alpha_n = \langle \varphi_n, f \rangle,</math>
which may be represented by the notation:
:<math>\alpha_n = \varphi_n^\dagger f, </math>
a form of the [[bra-ket notation]] of Dirac.<ref>

The development of this section in bra-ket notation is found in {{harv|Levin|2002|loc= Coordinate-space wave functions and completeness, pp.=109''ff''}}</ref> Adopting this notation, the expansion of ''f'' takes the [[Dyadic tensor|dyadic]] form:<ref>{{harvnb|Davis|Thomson|2000|loc=Perfect operators, p.344}}</ref>

:<math>f = \sum_{n=1}^\infty \varphi_n \left ( \varphi_n^\dagger f \right). </math>

Letting ''I'' denote the [[identity operator]] on the Hilbert space, the expression

:<math>I = \sum_{n=1}^\infty \varphi_n \varphi_n^\dagger, </math>

is called a [[Resolution_of_the_identity#Resolution_of_the_identity|resolution of the identity]]. When the Hilbert space is the space ''L''2(''D'') of square-integrable functions on a domain ''D'', the quantity:

:<math>\varphi_n \varphi_n^\dagger, </math>

is an integral operator, and the expression for ''f'' can be rewritten as:

:<math>f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi.</math>

The right-hand side converges to ''f'' in the ''L''2 sense. It need not hold in a pointwise sense, even when ''f'' is a continuous function. Nevertheless, it is common to abuse notation and write

:<math>f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, </math>

resulting in the representation of the delta function:<ref>{{harvnb|Davis|Thomson|2000|loc=Equation 8.9.11, p. 344}}</ref>

:<math>\delta(x-\xi) = \sum_{n=1}^\infty \varphi_n (x) \varphi_n^*(\xi). </math>

With a suitable [[rigged Hilbert space]] (Φ, ''L''2(''D''), Φ*) where Φ ⊂ ''L''2(''D'') contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φ''n''. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the [[Distribution_(mathematics)#Distributions|distribution]] sense.<ref>{{harvnb|de la Madrid|Bohm|Gadella|2002}}</ref>

===Infinitesimal delta functions===
[[Cauchy]] used an infinitesimal α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δα satisfying <math>\int F(x)\delta_\alpha(x) = F(0)</math> in a number of articles in 1827.<ref>See {{harvtxt|Laugwitz|1989}}.</ref> Cauchy defined an infinitesimal in Cours d'Analyse (1827) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and [[Lazare Carnot]]'s terminology.

Modern set-theoretic approaches allow one to define infinitesimals via the [[ultrapower]] construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable [[ultrafilter]]. The article by {{harvtxt|Yamashita|2007}} contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the [[hyperreal number|hyperreals]]. Here the Dirac delta can be given by an actual function, having the property that for every real function ''F'' one has <math>\int F(x)\delta_\alpha(x) = F(0)</math> as anticipated by Fourier and Cauchy.

==Dirac comb==
{{Main|Dirac comb}}
[[Image:Dirac comb.svg|thumb|A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of ''T'']]
A so-called uniform "pulse train" of Dirac delta measures, which is known as a [[Dirac comb]], or as the Shah distribution, creates a [[sampling (signal processing)|sampling]] function, often used in [[digital signal processing]] (DSP) and discrete time signal analysis. The Dirac comb is given as the [[infinite sum]], whose limit is understood in the distribution sense,

:<math>\Delta(x) = \sum_{n=-\infty}^\infty \delta(x-n),</math>

which is a sequence of point masses at each of the integers.

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if ''f'' is any [[Schwartz space|Schwartz function]], then the [[Wrapped distribution|periodization]] of ''f'' is given by the convolution
:<math>(f*\Delta)(x) = \sum_{n=-\infty}^\infty f(x-n).</math>
In particular,
:<math>(f*\Delta)^\wedge = \hat{f}\widehat{\Delta} = \hat{f}\Delta</math>
is precisely the [[Poisson summation formula]].<ref>{{harvnb|Córdoba|1988}}; {{harvnb|Hörmander|1983|loc=§7.2}}</ref>

==Sokhotski–Plemelj theorem==
The [[Sokhotski–Plemelj theorem]], important in quantum mechanics, relates the delta function to the distribution p.v.1/''x'', the [[Cauchy principal value]] of the function 1/''x'', defined by
:<math>\left\langle\operatorname{p.v.}\frac{1}{x}, \phi\right\rangle = \lim_{\varepsilon\to 0^+}\int_{|x|>\varepsilon} \frac{\phi(x)}{x}\,dx.</math>
Sokhatsky's formula states that<ref>{{harvnb|Vladimirov|1971|loc=§5.7}}</ref>
:<math>\lim_{\varepsilon\to 0^+} \frac{1}{x\pm i\varepsilon} = \operatorname{p.v.}\frac{1}{x} \mp i\pi\delta(x),</math>
Here the limit is understood in the distribution sense, that for all compactly supported smooth functions ''f'',
:<math>\lim_{\varepsilon\to 0^+} \int_{-\infty}^\infty\frac{f(x)}{x\pm i\varepsilon}\,dx = \mp i\pi f(0) + \lim_{\varepsilon\to 0^+} \int_{|x|>\varepsilon}\frac{f(x)}{x}\,dx.</math>

==Relationship to the Kronecker delta==
The [[Kronecker delta]] δij is the quantity defined by

:<math>\delta_{ij} = \begin{cases} 1 & i=j\\ 0 &i\not=j \end{cases} </math>

for all integers ''i'', ''j''. This function then satisfies the following analog of the sifting property: if <math>(a_i)_{i \in \mathbf{Z}}</math> is any [[Infinite_sequence#Doubly-infinite_sequences|doubly infinite sequence]], then

:<math>\sum_{i=-\infty}^\infty a_i \delta_{ik}=a_k.</math>

Similarly, for any real or complex valued continuous function ''f'' on '''R''', the Dirac delta satisfies the sifting property

:<math>\int_{-\infty}^\infty f(x)\delta(x-x_0)\,dx=f(x_0).</math>

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.<ref>{{harvnb|Hartmann|1997|loc=pp. 154–155}}</ref>

==Applications==
===Probability theory===
In [[probability theory]] and [[statistics]], the Dirac delta function is often used to represent a [[discrete distribution]], or a partially discrete, partially [[continuous distribution|continuous]] distribution, using a [[probability density function]] (which is normally used to represent fully continuous distributions). For example, the probability density function ''f''(''x'') of a discrete distribution consisting of points '''x''' = {''x''1, ..., ''xn''}, with corresponding probabilities ''p''1, ..., ''pn'', can be written as

:<math>f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

As another example, consider a distribution which 6/10 of the time returns a standard [[normal distribution]], and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete [[mixture distribution]]). The density function of this distribution can be written as

:<math>f(x) = 0.6 \, \frac {1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} + 0.4 \, \delta(x-3.5).</math>

The delta function is also used in a completely different way to represent the [[local time (mathematics)|local time]] of a [[diffusion process]] (like [[Brownian motion]]). The local time of a stochastic process ''B''(''t'') is given by
:<math>\ell(x,t) = \int_0^t \delta(x-B(s))\,ds</math>
and represents the amount of time that the process spends at the point ''x'' in the range of the process. More precisely, in one dimension this integral can be written
:<math>\ell(x,t) = \lim_{\varepsilon\to 0^+}\frac{1}{2\varepsilon}\int_0^t \mathbf{1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds</math>
where '''1'''[''x''−ε, ''x''+ε] is the [[indicator function]] of the interval [''x''−ε, ''x''+ε].

===Quantum mechanics===
We give an example of how the delta function is expedient in [[quantum mechanics]]. The [[wave function]] of a particle gives the probability amplitude of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space ''L''2 of [[square-integrable function]]s, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set {φ''n''} of wave functions is orthonormal if they are normalized by

:<math>\langle\phi_n|\phi_m\rangle = \delta_{nm}</math>

where δ here refers to the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function ''ψ'' can be expressed as a combination of the φ''n'':

: <math> \psi = \sum c_n \phi_n, </math>

with <math> c_n = \langle \phi_n | \psi \rangle </math>. Complete orthonormal systems of wave functions appear naturally as the [[eigenfunction]]s of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (of a [[bound state|bound system]]) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the [[spectrum]] of the Hamiltonian. In [[bra-ket notation]], as [[#Resolutions of the identity|above]], this equality implies the resolution of the identity:

: <math>I = \sum |\phi_n\rangle\langle\phi_n|.</math>

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an [[observable]] may be continuous rather than discrete. An example is the [[position operator|position observable]], ''Qψ''(''x'') = ''x''ψ(''x''). The spectrum of the position (in one dimension) is the entire real line, and is called a [[continuous spectrum]]. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well: that is, to replace the Hilbert space of quantum mechanics by an appropriate [[rigged Hilbert space]].<ref>{{harvnb|Isham|1995|loc=§6.2}}</ref> In this context, the position operator has a complete set of eigen-distributions, labeled by the points ''y'' of the real line, given by

:<math>\phi_y(x) = \delta(x-y).\;</math>

The eigenfunctions of position are denoted by <math>\phi_y = |y\rangle</math> in Dirac notation, and are known as position eigenstates.

Similar considerations apply to the eigenstates of the [[momentum operator]], or indeed any other self-adjoint [[unbounded operator]] ''P'' on the Hilbert space, provided the spectrum of ''P'' is continuous and there are no degenerate eigenvalues. In that case, there is a set Ω of real numbers (the spectrum), and a collection φ''y'' of distributions indexed by the elements of Ω, such that

:<math>P\phi_y = y\phi_y.\;</math>

That is, φ''y'' are the eigenvectors of ''P''. If the eigenvectors are normalized so that

:<math>\langle \phi_y,\phi_{y'}\rangle = \delta(y-y')</math>

in the distribution sense, then for any test function ψ,

: <math> \psi(x) = \int_\Omega c(y) \phi_y(x) \, dy</math>

where

: <math>c(y) = \langle \psi, \phi_y \rangle.</math>

That is, as in the discrete case, there is a resolution of the identity

:<math>I = \int_\Omega |\phi_y\rangle\, \langle\phi_y|\,dy</math>

where the operator-valued integral is again understood in the weak sense. If the spectrum of ''P'' has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum ''and'' an integral over the continuous spectrum.

The delta function also has many more specialized applications in quantum mechanics, such as the [[delta potential]] models for a single and double potential well.

===Structural mechanics===
The delta function can be used in [[structural mechanics]] to describe transient loads or point loads acting on structures. The governing equation of a simple [[Harmonic oscillator|mass–spring system]] excited by a sudden force [[impulse (physics)|impulse]] ''I'' at time ''t'' = 0 can be written

:<math>m \frac{\mathrm{d}^2 \xi}{\mathrm{d} t^2} + k \xi = I \delta(t),</math>

where ''m'' is the mass, ξ the deflection and ''k'' the [[spring constant]].

As another example, the equation governing the static deflection of a slender [[beam (structure)|beam]] is, according to [[Euler-Bernoulli beam equation|Euler-Bernoulli theory]],

:<math>EI \frac{\mathrm{d}^4 w}{\mathrm{d} x^4} = q(x),\,</math>

where ''EI'' is the [[bending stiffness]] of the beam, ''w'' the [[deflection (engineering)|deflection]], ''x'' the spatial coordinate and ''q''(''x'') the load distribution. If a beam is loaded by a point force ''F'' at ''x'' = ''x''0, the load distribution is written

:<math>q(x) = F \delta(x-x_0).\,</math>

As integration of the delta function results in the [[Heaviside step function]], it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise [[polynomial]]s.

Also a point [[bending moment|moment]] acting on a beam can be described by delta functions. Consider two opposing point forces ''F'' at a distance ''d'' apart. They then produce a moment ''M'' = ''Fd'' acting on the beam. Now, let the distance ''d'' approach the [[Limit of a function|limit]] zero, while ''M'' is kept constant. The load distribution, assuming a clockwise moment acting at ''x'' = 0, is written

:<math>\begin{align}
q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\
&= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\
&= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\
&= M \delta'(x).
\end{align}</math>

Point moments can thus be represented by the [[derivative]] of the delta function. Integration of the beam equation again results in piecewise [[polynomial]] deflection.

==See also==
*[[Atom (measure theory)]]
*[[Delta potential]]
*[[Dirac measure]]
*[[Fundamental solution]]
*[[Green's function]]
*[[Laplacian of the indicator]]

==Notes==
{{clear}}
{{Reflist|colwidth=30em}}

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*{{citation|last=Lange|first=Rutger-Jan|year=2012|publisher=Springer|title=Potential theory, path integrals and the Laplacian of the indicator|journal=Journal of High Energy Physics|volume=2012|pages=29–30|url=http://link.springer.com/article/10.1007%2FJHEP11(2012)032|issue=11|bibcode=2012JHEP...11..032L|doi=10.1007/JHEP11(2012)032|arxiv = 1302.0864 }}.
*{{citation|doi=10.1007/BF00329867|authorlink=Detlef Laugwitz|last=Laugwitz|first=D.|year=1989|title=Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820|journal=Arch. Hist. Exact Sci.|volume=39|issue=3|pages=195–245}}.
*{{Citation |title=An introduction to quantum theory |last=Levin |first=Frank S. |year=2002 |url=http://books.google.com/?id=oc64f4EspFgC&pg=PA109 |pages=109''ff'' |chapter=Coordinate-space wave functions and completeness |isbn=0-521-59841-9 |publisher=Cambridge University Press|ref =harv}}
*{{citation|last1=Li|first1=Y. T.|last2=Wong|first2=R.|mr=2373214|title=Integral and series representations of the Dirac delta function|journal=Commun. Pure Appl. Anal.|volume=7|year=2008|issue=2|doi=10.3934/cpaa.2008.7.229|pages=229–247}}.
*{{citation|doi=10.1002/1521-3978(200203)50:2<185::AID-PROP185>3.0.CO;2-S|first1=R.|last1=de la Madrid|first2=A.|last2=Bohm|first3=M.|last3=Gadella|title=Rigged Hilbert Space Treatment of Continuous Spectrum|arxiv=quant-ph/0109154|journal=Fortschr. Phys.|volume=50|year=2002|pages=185–216|issue=2|bibcode = 2002ForPh..50..185D }}.
*{{citation|last=McMahon|first=D.|title=Quantum Mechanics Demystified, A Self-Teaching Guide|url=http://www.mhprofessional.com/product.php?isbn=0071455469&cat=&promocode=|accessdate=2008-03-17|series=Demystified Series|date=2005-11-22|publisher=McGraw-Hill|location=New York|isbn=0-07-145546-9|doi=10.1036/0071455469|pages=108|chapter=An Introduction to State Space}}.
*{{Citation | last1=van der Pol | first1=Balth. | last2=Bremmer | first2=H. | title=Operational calculus | publisher=Chelsea Publishing Co. | location=New York | edition=3rd | isbn=978-0-8284-0327-6 | mr=904873 | year=1987}}.
* {{citation|first=W.|last=Rudin|authorlink=Walter Rudin|title=Functional Analysis|edition=2nd|publisher=McGraw-Hill|year=1991|isbn=0-07-054236-8}}.
* {{citation|first2=Olivier|last2=Vallée|first1=Manuel|last1=Soares|year=2004|title=Airy functions and applications to physics|publisher=Imperial College Press|publication-place=London}}.
*{{Citation|last1=Saichev |first1=A I |last2=Woyczyński |first2=Wojbor Andrzej |title=Distributions in the Physical and Engineering Sciences: Distributional and fractal calculus, integral transforms, and wavelets |year=1997 |isbn=0-8176-3924-1 |publisher=Birkhäuser |url=http://books.google.com/?id=42I7huO-hiYC&pg=PA3|chapter=Chapter1: Basic definitions and operations }}
* {{citation|first=L.|last=Schwartz|authorlink=Laurent Schwartz|title=Théorie des distributions|volume=1|publisher=Hermann|year=1950}}.
* {{citation|first=L.|last=Schwartz|authorlink=Laurent Schwartz|title=Théorie des distributions|volume=2|publisher=Hermann|year=1951}}.
* {{citation|first1=Elias|last1=Stein|authorlink1=Elias Stein|first2=Guido|last2=Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=0-691-08078-X}}.
* {{citation|first=R.|last=Strichartz|year=1994|title=A Guide to Distribution Theory and Fourier Transforms|publisher=CRC Press|isbn=0-8493-8273-4}}.
* {{citation|first=V. S.|last=Vladimirov|title=Equations of mathematical physics|publisher=Marcel Dekker|year=1971|isbn=0-8247-1713-9}}.
* {{MathWorld|urlname=DeltaFunction|title=Delta Function}}
* {{citation|last=Yamashita |first=H. |year=2006 |title= Pointwise analysis of scalar fields: A nonstandard approach |volume=47 |issue =9 |pages=092301|journal=[[Journal of Mathematical Physics]]|doi=10.1063/1.2339017|bibcode = 2006JMP....47i2301Y }}
* {{citation|last=Yamashita |first=H. |year=2007 |title= Comment on "Pointwise analysis of scalar fields: A nonstandard approach" [J. Math. Phys. 47, 092301 (2006)] |volume=48 |issue =8 |pages=084101|journal=[[Journal of Mathematical Physics]]|doi=10.1063/1.2771422|bibcode = 2007JMP....48h4101Y }}

==External links==
*{{springer|title=Delta-function|id=p/d030950}}
*[http://www.khanacademy.org/video/dirac-delta-function KhanAcademy.org video lesson]
*[http://www.physicsforums.com/showthread.php?t=73447 The Dirac Delta function], a tutorial on the Dirac delta function.
*[http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-23-use-with-impulse-inputs Video Lectures – Lecture 23], a lecture by [[Arthur Mattuck]].
*[http://planetmath.org/encyclopedia/DiracDeltaFunction.html Dirac Delta Function] on [[PlanetMath]]
*[http://www.osaka-kyoiku.ac.jp/~ashino/pdf/chinaproceedings.pdf The Dirac delta measure is a hyperfunction]
*[http://www.ing-mat.udec.cl/~rodolfo/Papers/BGR-3.pdf We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure]
*[http://www.mathematik.uni-muenchen.de/~lerdos/WS04/FA/content.html Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure.]

{{ProbDistributions|miscellaneous}}
{{Infinitesimal navbox}}

{{good article}}

{{DEFAULTSORT:Dirac Delta Function}}
[[Category:Fourier analysis]]
[[Category:Generalized functions]]
[[Category:Measure theory]]
[[Category:Digital signal processing]]
[[Category:Paul Dirac|Delta function]]

Insertion device

2013-12-04T02:52:13Z

193.190.253.144:

{{Bayesian statistics}}
'''Bayesian spam filtering''' ({{IPAc-en|ˈ|b|eɪ|z|i|ə|n}} {{respell|BAY|zee-ən}}; after Rev. [[Thomas Bayes]]) is a [[statistics|statistical]] [[scientific technique|technique]] of [[e-mail filtering]]. In its basic form, it makes use of a [[naive Bayes classifier]] on [[bag of words]] features to identify [[Spam (electronic)|spam]] e-mail, an approach commonly used in [[Document classification|text classification]].

Naive Bayes classifiers work by correlating the use of tokens (typically words, or sometimes other things), with spam and non-spam e-mails and then using [[Bayesian inference]] to calculate a probability that an email is or is not spam.

Naive Bayes spam filtering is a baseline technique for dealing with spam that can tailor itself to the email needs of individual users and give low [[false positive]] spam detection rates that are generally acceptable to users. It is one of the oldest ways of doing spam filtering, with roots in the 1990s.

==History==
The first known mail-filtering [[computer program|program]] to use a naive Bayes classifier was Jason Rennie's ifile program, released in 1996. The program was used to sort mail into [[Directory (file systems)|folders]].<ref>{{cite web|url=http://people.csail.mit.edu/jrennie/ifile/old/README-0.1A|author=Jason Rennie|title=ifile|year=1996}}</ref> The first scholarly publication on Bayesian spam filtering was by Sahami et al. in 1998.<ref>{{cite web|url=http://robotics.stanford.edu/users/sahami/papers-dir/spam.pdf|author=M. Sahami, S. Dumais, D. Heckerman, E. Horvitz|title=A Bayesian approach to filtering junk e-mail|publisher=AAAI'98 Workshop on Learning for Text Categorization|year=1998}}</ref> That work was soon thereafter deployed in commercial spam filters.{{Citation needed|date=September 2010}} However, in 2002 [[Paul Graham (computer programmer)|Paul Graham]] greatly decreased the false positive rate, so that it could be used on its own as a single spam filter.<ref>Paul Graham (2003), [http://www.paulgraham.com/better.html Better Bayesian filtering]</ref><ref>Brian Livingston (2002), [http://www.infoworld.com/t/business/paul-graham-provides-stunning-answer-spam-e-mails-295 Paul Graham provides stunning answer to spam e-mails]</ref>

Variants of the basic technique have been implemented in a number of research works and commercial [[Computer software|software]] products.<ref>{{cite web|url=http://kb.mozillazine.org/Junk_Mail_Controls|title=Junk Mail Controls|publisher=MozillaZine|date=November 2009}}</ref> Many modern mail [[Client (computing)|clients]] implement Bayesian spam filtering. Users can also install separate [[E-mail filtering|email filtering programs]]. [[Server-side]] email filters, such as [[CRM114 (program)|CRM114]], [[DSPAM]], [[SpamAssassin]],<ref name=twsSep14yy>{{cite web
|title= Installation
|publisher= ''Ubuntu manuals''
|quote= Gary Robinson’s f(x) and combining algorithms, as used in SpamAssassin
|date= 2010-09-18
|url= http://manpages.ubuntu.com/manpages/gutsy/man1/sa-learn.1p.html
|accessdate= 2010-09-18
| archiveurl= http://web.archive.org/web/20100929165032/http://manpages.ubuntu.com/manpages/gutsy/man1/sa-learn.1p.html| archivedate= 29 September 2010 | deadurl= no}}</ref> [[SpamBayes]],<ref name=twsSep2>{{Cite news
|title= Background Reading
|publisher= ''SpamBayes project''
|quote= Sharpen your pencils, this is the mathematical background (such as it is).* The paper that started the ball rolling: Paul Graham's A Plan for Spam.* Gary Robinson has an interesting essay suggesting some improvements to Graham's original approach.* Gary Robinson's Linux Journal article discussed using the chi squared distribution.
|date= 2010-09-18
|url= http://spambayes.sourceforge.net/background.html
|accessdate= 2010-09-18
| archiveurl= http://web.archive.org/web/20100906031341/http://spambayes.sourceforge.net/background.html| archivedate= 6 September 2010 | deadurl= no}}</ref> [[Bogofilter]] and [[Anti-Spam SMTP Proxy|ASSP]], make use of Bayesian spam filtering techniques, and the functionality is sometimes embedded within [[mail server]] software itself.

==Process==
Particular words have particular [[probability|probabilities]] of occurring in spam email and in legitimate email. For instance, most email users will frequently encounter the word "[[Viagra]]" in spam email, but will seldom see it in other email. The filter doesn't know these probabilities in advance, and must first be trained so it can build them up. To train the filter, the user must manually indicate whether a new email is spam or not. For all words in each training email, the filter will adjust the probabilities that each word will appear in spam or legitimate email in its database. For instance, Bayesian spam filters will typically have learned a very high spam probability for the words "Viagra" and "refinance", but a very low spam probability for words seen only in legitimate email, such as the names of friends and family members.

After training, the word probabilities (also known as [[likelihood function]]s) are used to compute the probability that an email with a particular set of words in it belongs to either category. Each word in the email contributes to the email's spam probability, or only the most interesting words. This contribution is called the [[posterior probability]] and is computed using [[Bayes' theorem]]. Then, the email's spam probability is computed over all words in the email, and if the total exceeds a certain threshold (say 95%), the filter will mark the email as a spam.

As in any other [[spam filtering]] technique, email marked as spam can then be automatically moved to a "Junk" email folder, or even deleted outright. Some software implement [[quarantine]] mechanisms that define a time frame during which the user is allowed to review the software's decision.

The initial training can usually be refined when wrong judgements from the software are identified (false positives or false negatives). That allows the software to dynamically adapt to the ever evolving nature of spam.

Some spam filters combine the results of both Bayesian spam filtering and other [[metaheuristic|heuristics]] (pre-defined rules about the contents, looking at the message's envelope, etc.), resulting in even higher filtering accuracy, sometimes at the cost of adaptiveness.

==Mathematical foundation==
Bayesian [[email filter]]s utilize [[Bayes' theorem]]. Bayes' theorem is used several times in the context of spam:
* a first time, to compute the probability that the message is spam, knowing that a given word appears in this message;
* a second time, to compute the probability that the message is spam, taking into consideration all of its words (or a relevant subset of them);
* sometimes a third time, to deal with rare words.

===Computing the probability that a message containing a given word is spam===
Let's suppose the suspected message contains the word "[[replica]]". Most people who are used to receiving e-mail know that this message is likely to be spam, more precisely a proposal to sell counterfeit copies of well-known brands of watches. The spam detection software, however, does not "know" such facts; all it can do is compute probabilities.

The formula used by the software to determine that is derived from [[Bayes' theorem]]

:<math>\Pr(S|W) = \frac{\Pr(W|S) \cdot \Pr(S)}{\Pr(W|S) \cdot \Pr(S) + \Pr(W|H) \cdot \Pr(H)}</math>

where:

* <math>\Pr(S|W)</math> is the probability that a message is a spam, knowing that the word "replica" is in it;
* <math>\Pr(S)</math> is the overall probability that any given message is spam;
* <math>\Pr(W|S)</math> is the probability that the word "replica" appears in spam messages;
* <math>\Pr(H)</math> is the overall probability that any given message is not spam (is "ham");
* <math>\Pr(W|H)</math> is the probability that the word "replica" appears in ham messages.

(For a full demonstration, see [[Bayes' theorem#Extended form]].)

===The spamicity of a word===
Recent statistics<ref>{{cite web|url=http://eval.symantec.com/mktginfo/enterprise/other_resources/b-state_of_spam_report_09-2009.en-us.pdf|author=Dylan Mors and Dermot Harnett|title=State of Spam, a Monthly Report - Report #33|year=2009}}</ref> show that the current probability of any message being spam is 80%, at the very least:
:<math> \Pr(S) = 0.8 ; \Pr(H) = 0.2</math>

However, most bayesian spam detection software makes the assumption that there is no ''a priori'' reason for any incoming message to be spam rather than ham, and considers both cases to have equal probabilities of 50%:{{citation needed|date=July 2012}}

:<math> \Pr(S) = 0.5 ; \Pr(H) = 0.5</math>

The filters that use this hypothesis are said to be "not biased", meaning that they have no prejudice regarding the incoming email. This assumption permits simplifying the general formula to:

:<math>\Pr(S|W) = \frac{\Pr(W|S)}{\Pr(W|S) + \Pr(W|H)}</math>

This quantity is called "spamicity" (or "spaminess") of the word "replica", and can be computed. The number <math>\Pr(W|S)</math> used in this formula is approximated to the frequency of messages containing "replica" in the messages identified as spam during the learning phase. Similarly, <math>\Pr(W|H)</math> is approximated to the frequency of messages containing "replica" in the messages identified as ham during the learning phase. For these approximations to make sense, the set of learned messages needs to be big and representative enough. It is also advisable that the learned set of messages conforms to the 50% hypothesis about repartition between spam and ham, i.e. that the datasets of spam and ham are of same size.<ref>Process Software, [http://www.process.com/precisemail/bayesian_filtering.htm Introduction to Bayesian Filtering]</ref>

Of course, determining whether a message is spam or ham based only on the presence of the word "replica" is error-prone, which is why bayesian spam software tries to consider several words and combine their spamicities to determine a message's overall probability of being spam.

===Combining individual probabilities===
Most bayesian spam filtering algorithms are based on formulas that are strictly valid (from a probabilistic standpoint) only if the words present in the message are [[Statistical independence|independent events]]. This condition is not generally satisfied (for example, in natural languages like English the probability of finding an adjective is affected by the probability of having a noun), but it is a useful idealization, especially since the statistical correlations between individual words are usually not known. On this basis, one can derive the following formula from Bayes' theorem:<ref>{{cite web|url=http://www.mathpages.com/home/kmath267.htm|title=Combining probabilities}} at MathPages</ref>

:<math>p = \frac{p_1 p_2 \cdots p_N}{p_1 p_2 \cdots p_N + (1 - p_1)(1 - p_2) \cdots (1 - p_N)}</math>

where:
* <math>p</math> is the probability that the suspect message is spam;
* <math>p_1</math> is the probability <math>p(S|W_1)</math> that it is a spam knowing it contains a first word (for example "replica");
* <math>p_2</math> is the probability <math>p(S|W_2)</math> that it is a spam knowing it contains a second word (for example "watches");
* etc...
* <math>p_N</math> is the probability <math>p(S|W_N)</math> that it is a spam knowing it contains an ''N''th word (for example "home").

This is the formula referenced by Paul Graham in his 2002 article. Some early commentators stated that "Graham pulled his formulas out of thin air",<ref>http://mail.python.org/pipermail/python-dev/2002-August/028216.html Tim Peter's comment on the algorithm used by Graham</ref> but Graham had actually referenced his source,<ref>{{cite web|url=http://www.paulgraham.com/naivebayes.html|title=Graham's web page referencing the MathPages article for the probability formula used in his spam algorithm.}}</ref> which included a detailed explanation of the formula, and the idealizations on which it is based.

Spam filtering software based on this formula is sometimes referred to as a [[naive Bayes classifier]]. The result ''p'' is typically compared to a given threshold to decide whether the message is spam or not. If ''p'' is lower than the threshold, the message is considered as likely ham, otherwise it is considered as likely spam.

===Other expression of the formula for combining individual probabilities===

Usually ''p'' is not directly computed using the above formula due to [[Arithmetic underflow|floating-point underflow]]. Instead, ''p'' can be computed in the log domain by rewriting the original equation as follows:

:<math> \frac{1}{p} - 1 = \frac{(1-p_1)(1-p_2)\dots(1-p_n)}{p_1 p_2 \dots p_n} </math>

Taking logs on both sides:

:<math> \ln \left ( \frac{1}{p} - 1 \right ) = \sum_{i=1}^N \left[ \ln(1-p_i) - \ln p_i \right]</math>

Let <math>\eta = \sum_{i=1}^N \left[ \ln(1-p_i) -\ln p_i \right] </math>. Therefore,

:<math> \frac{1}{p} - 1 = e^\eta </math>

Hence the alternate formula for computing the combined probability:

:<math> p = \frac{1}{1 + e^\eta} </math>

===Dealing with rare words===
In the case a word has never been met during the learning phase, both the numerator and the denominator are equal to zero, both in the general formula and in the spamicity formula. The software can decide to discard such words for which there is no information available.

More generally, the words that were encountered only a few times during the learning phase cause a problem, because it would be an error to trust blindly the information they provide. A simple solution is to simply avoid taking such unreliable words into account as well.

Applying again Bayes' theorem, and assuming the classification between spam and ham of the emails containing a given word ("replica") is a [[random variable]] with [[beta distribution]], some programs decide to use a corrected probability:

:<math>\Pr'(S|W) = \frac{s \cdot \Pr(S) + n \cdot \Pr(S|W)}{s + n }</math>

where:
*<math>\Pr'(S|W)</math> is the corrected probability for the message to be spam, knowing that it contains a given word ;
* <math>s</math> is the ''strength'' we give to background information about incoming spam ;
* <math>\Pr(S)</math> is the probability of any incoming message to be spam ;
* <math>n</math> is the number of occurrences of this word during the learning phase ;
* <math>\Pr(S|W)</math> is the spamicity of this word.

(Demonstration:<ref>{{cite web|url=http://www.linuxjournal.com/article/6467|publisher=Linux Journal|author=[[Gary Robinson]]|title=A statistical approach to the spam problem|year=2003}}</ref>)

This corrected probability is used instead of the spamicity in the combining formula.

<math>\Pr(S)</math> can again be taken equal to 0.5, to avoid being too suspicious about incoming email. 3 is a good value for ''s'', meaning that the learned corpus must contain more than 3 messages with that word to put more confidence in the spamicity value than in the default value.

This formula can be extended to the case where ''n'' is equal to zero (and where the spamicity is not defined), and evaluates in this case to <math>Pr(S)</math>.

===Other heuristics===
"Neutral" words like "the", "a", "some", or "is" (in English), or their equivalents in other languages, can be ignored. More generally, some bayesian filtering filters simply ignore all the words which have a spamicity next to 0.5, as they bring little to a good decision. The words taken into consideration are those whose spamicity is next to 0.0 (distinctive signs of legitimate messages), or next to 1.0 (distinctive signs of spam). A method can be for example to keep only those ten words, in the examined message, which have the greatest [[absolute value]] |0.5 − ''pI''|.

Some software products take into account the fact that a given word appears several times in the examined message,<ref>{{cite web|url=http://spamprobe.sourceforge.net/paper.html|author=Brian Burton|title=SpamProbe - Bayesian Spam Filtering Tweaks|year=2003}}</ref> others don't.

Some software products use ''patterns'' (sequences of words) instead of isolated natural languages words.<ref>{{cite web|url=http://bnr.nuclearelephant.com/l|author=Jonathan A. Zdziarski|title=Bayesian Noise Reduction: Contextual Symmetry Logic Utilizing Pattern Consistency Analysis|year=2004}}</ref> For example, with a "context window" of four words, they compute the spamicity of "Viagra is good for", instead of computing the spamicities of "Viagra", "is", "good", and "for". This method gives more sensitivity to context and eliminates the [[Bayesian noise]] better, at the expense of a bigger database.

===Mixed methods===
There are other ways of combining individual probabilities for different words than using the "naive" approach. These methods differ from it on the assumptions they make on the statistical properties of the input data. These different hypotheses result in radically different formulas for combining the individual probabilities.

For example, assuming the individual probabilities follow a [[chi-squared distribution|chi-squared]] distribution with 2''N'' degrees of freedom, one could use the formula:

:<math>p = C^{-1}(-2 \ln(p_1 p_2 \cdots p_N), 2N) \, </math>

where ''C''−1 is the [[Inverse-chi-squared distribution|inverse of the chi-squared function]].

Individual probabilities can be combined with the techniques of the [[Markovian discrimination]] too.

==Discussion==

===Advantages===
{{disputed-section|date=May 2013}}
One of the main advantages{{citation needed|date=May 2013}} of Bayesian spam filtering is that it can be trained on a per-user basis.

The spam that a user receives is often related to the online user's activities. For example, a user may have been subscribed to an online newsletter that the user considers to be spam. This online newsletter is likely to contain words that are common to all newsletters, such as the name of the newsletter and its originating email address. A Bayesian spam filter will eventually assign a higher probability based on the user's specific patterns.

The legitimate e-mails a user receives will tend to be different. For example, in a corporate environment, the company name and the names of clients or customers will be mentioned often. The filter will assign a lower spam probability to emails containing those names.

The word probabilities are unique to each user and can evolve over time with corrective training whenever the filter incorrectly classifies an email. As a result, Bayesian spam filtering accuracy after training is often superior to pre-defined rules.

It can perform particularly well in avoiding false positives,{{citation needed|date=May 2013}} where legitimate email is incorrectly classified as spam. For example, if the email contains the word "Nigeria", which is frequently used in [[Advance fee fraud]] spam, a pre-defined rules filter might reject it outright. A Bayesian filter would mark the word "Nigeria" as a probable spam word, but would take into account other important words that usually indicate legitimate e-mail. For example, the name of a spouse may strongly indicate the e-mail is not spam, which could overcome the use of the word "Nigeria."

===Disadvantages===
Depending on the implementation, Bayesian spam filtering may be susceptible to [[Bayesian poisoning]], a technique used by spammers in an attempt to degrade the effectiveness of spam filters that rely on Bayesian filtering. A spammer practicing Bayesian poisoning will send out emails with large amounts of legitimate text (gathered from legitimate news or literary sources). [[e-mail spam|Spammer]] tactics include insertion of random innocuous words that are not normally associated with spam, thereby decreasing the email's spam score, making it more likely to slip past a Bayesian spam filter. However with (for example) Paul Graham's scheme only the most significant probabilities are used, so that padding the text out with non-spam-related words does not affect the detection probability significantly.

Words that normally appear in large quantities in spam may also be transformed by spammers. For example, «Viagra» would be replaced with «Viaagra» or «V!agra» in the spam message. The recipient of the message can still read the changed words, but each of these words is met more rarely by the Bayesian filter, which hinders its learning process. As a general rule, this spamming technique does not work very well, because the derived words end up recognized by the filter just like the normal ones.<ref>Paul Graham (2002), [http://www.paulgraham.com/spam.html A Plan for Spam]</ref>

Another technique used to try to defeat Bayesian spam filters is to replace text with pictures, either directly included or linked. The whole text of the message, or some part of it, is replaced with a picture where the same text is "drawn". The spam filter is usually unable to analyze this picture, which would contain the sensitive words like «Viagra». However, since many mail clients disable the display of linked pictures for security reasons, the spammer sending links to distant pictures might reach fewer targets. Also, a picture's size in bytes is bigger than the equivalent text's size, so the spammer needs more bandwidth to send messages directly including pictures. Some filters are more inclined to decide that a message is spam if it has mostly graphical contents. Finally, a probably more efficient solution has been proposed by Google and is used by its [[Gmail]] email system, performing an [[Optical character recognition|OCR (Optical Character Recognition)]] to every mid to large size image, analyzing the text inside.<ref>{{cite web|url=http://www.google.com/mail/help/fightspam/spamexplained.html|title=Gmail uses Google's innovative technology to keep spam out of your inbox}}</ref>

==General applications of Bayesian filtering==
While Bayesian filtering is used widely to identify spam email, the technique can classify (or "cluster") almost any sort of data. It has uses in science, medicine, and engineering. One example is a general purpose classification program called [http://ti.arc.nasa.gov/tech/rse/synthesis-projects-applications/autoclass/ AutoClass] which was originally used to classify stars according to spectral characteristics that were otherwise too subtle to notice. There is recent speculation that even the brain uses Bayesian methods to classify sensory stimuli and decide on behavioral responses.<ref>[http://www.bcs.rochester.edu/people/alex/pub/articles/KnillPougetTINS04.pdf Trends in Neuroscience, 27(12):712-9, 2004] (pdf)</ref>

==See also==
* [[Bayesian poisoning]]
* [[Bayesian programming]]
* [[Bayesian inference]]
* [[Bayes's theorem]]
* [[Email filtering]]
* [[Markovian discrimination]]
* [[Naive Bayes classifier]]
* [[Recursive Bayesian estimation]]
* [[Stopping e-mail abuse]]

==References==
<references />

==External links==
* Guide to Bayesian spam filters: [http://lwn.net/Articles/172491/ part 1], [http://lwn.net/Articles/173910/ part 2].
* [http://radio.weblogs.com/0101454/stories/2002/09/16/spamDetection.html Gary Robinson's spam blog]

{{DEFAULTSORT:Bayesian Spam Filtering}}
[[Category:Applications of Bayesian inference|Spam filtering]]
[[Category:Estimation theory]]
[[Category:Spam filtering]]

Laue equations

2013-05-27T14:00:39Z

193.190.253.150: Whence -> Hence

{{Probability distribution |
name =Skew Normal|
type =density|
pdf_image =[[Image:Skew normal densities.svg|325px|Probability density plots of skew normal distributions]]|
cdf_image =[[Image:Skew normal cdfs.svg|325px|Cumulative distribution function plots of skew normal distributions]]|
parameters =<math>\xi \,</math> [[location parameter|location]] ([[real number|real]]) <math>\omega \,</math> [[scale parameter|scale]] (positive, [[real number|real]]) <math>\alpha \,</math> [[shape parameter|shape]] ([[real number|real]])|
support =<math>x \in (-\infty; +\infty)\!</math>|
pdf = <math>\frac{1}{\omega\pi} e^{-\frac{(x-\xi)^2}{2\omega^2}} \int_{-\infty}^{\alpha\left(\frac{x-\xi}{\omega}\right)} e^{-\frac{t^2}{2}}\ dt</math>|
cdf =<math>\Phi\left(\frac{x-\xi}{\omega}\right)-2T\left(\frac{x-\xi}{\omega},\alpha\right)</math> <math>T(h,a)</math> is [[Owen's T function]]|
mean =<math>\xi + \omega\delta\sqrt{\frac{2}{\pi}}</math> where <math>\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}</math>|
median =|
mode =|
variance =<math>\omega^2\left(1 - \frac{2\delta^2}{\pi}\right)</math>|
skewness =<math>\gamma_3 = \frac{4-\pi}{2} \frac{\left(\delta\sqrt{2/\pi}\right)^3}{ \left(1-2\delta^2/\pi\right)^{3/2}}</math>|
kurtosis =<math>2(\pi - 3)\frac{\left(\delta\sqrt{2/\pi}\right)^4}{\left(1-2\delta^2/\pi\right)^2}</math>|
entropy =|
mgf =<math>M_X\left(t\right)=2\exp\left(\xi t+\frac{\omega^2t^2}{2}\right)\Phi\left(\omega\delta t\right)</math>|
char =|
}}

In [[probability theory]] and [[statistics]], the '''skew normal distribution''' is a [[continuous probability distribution]] that generalises the [[normal distribution]] to allow for non-zero [[skewness]].

== Definition ==

Let <math>\phi(x)</math> denote the standard normal [[probability density function]]
:<math>\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}</math>
with the [[cumulative distribution function]] given by
:<math>\Phi(x) = \int_{-\infty}^{x} \phi(t)\ dt = \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right)\right]</math>

where '''erf''' is the [[error function]]. Then the probability density function of the skew-normal distribution with parameter α is given by
:<math>f(x) = 2\phi(x)\Phi(\alpha x). \,</math>

This distribution was first introduced by O'Hagan and Leonard (1976).

To add [[location parameter|location]] and [[scale parameter|scale]] parameters to this, one makes the usual transform <math>x\rightarrow\frac{x-\xi}{\omega}</math>. One can verify that the normal distribution is recovered when <math>\alpha = 0</math>, and that the absolute value of the [[skewness]] increases as the absolute value of <math>\alpha</math> increases. The distribution is right skewed if <math>\alpha>0</math> and is left skewed if <math>\alpha<0</math>. The probability density function with location <math>\xi</math>, scale <math>\omega</math>, and parameter <math>\alpha</math> becomes
:<math>f(x) = \frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha \left(\frac{x-\xi}{\omega}\right)\right). \,</math>
Note, however, that the skewness of the distribution is limited to the interval <math>(-1,1)</math>.

==Estimation==

[[Maximum likelihood]] estimates for <math>\xi</math>, <math>\omega</math>, and <math>\alpha</math> can be computed numerically, but no closed-form expression for the estimates is available unless <math>\alpha=0</math>. If a closed-form expression is needed, the [[Method of moments (statistics)|method of moments]] can be applied to estimate <math>\alpha</math> from the sample skew, by inverting the skewness equation. This yields the estimate

:<math>|\delta| = \sqrt{\frac{\pi}{2} \frac{ |\hat{\gamma}_3|^{\frac{2}{3}} }{|\hat{\gamma}_3|^{\frac{2}{3}}+((4-\pi)/2)^\frac{2}{3}}}</math>

where <math>\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}</math>, and <math>\hat{\gamma}_3</math> is the sample skew. The sign of <math>\delta</math> is the same as the sign of <math>\hat{\gamma}_3</math>. Consequently, <math>\hat{\alpha} = \delta/\sqrt{1-\delta^2}</math>.

The maximum (theoretical) skewness is obtained by setting <math>{\delta = 1}</math> in the skewness equation, giving <math>\gamma_3 \approx 0.9952717</math>. However it is possible that the sample skewness is larger, and then <math>\alpha</math> cannot be determined from these equations. When using the method of moments in an automatic fashion, for example to give starting values for maximum likelihood iteration, one should therefore let (for example) <math>|\hat{\gamma}_3| = \min(0.99, |(1/n)\sum{((x_i-\bar{x})/s)^3}|)</math>.

==See also==
* [[Normal distribution]]
* [[Generalized normal distribution]]
* [[Log-normal distribution]]
* [[Skewness]]

==References==
* {{cite journal |last=Azzalini |first=A. |authorlink= |coauthors= |year=1985 |title=A class of distributions which includes the normal ones|journal=Scandinavian Journal of Statistics |volume=12 |issue= |pages=171–178}}

* O'Hagan, A. and Leonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints. Biometrika, 63, 201-202.

==External links==
* [http://azzalini.stat.unipd.it/SN/Intro/intro.html A very brief introduction to the skew-normal distribution]
* [http://azzalini.stat.unipd.it/SN/ The Skew-Normal Probability Distribution (and related distributions, such as the skew-t)]
* [http://people.sc.fsu.edu/~burkardt/cpp_src/owens/owens.html OWENS: Owen's T Function]
* [http://dahoiv.net/master/index.html Closed-skew Distributions - Simulation, Inversion and Parameter Estimation]

{{ProbDistributions|continuous-infinite}}
{{Statistics|hide}}

{{DEFAULTSORT:Skew Normal Distribution}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Volume of fluid method

2012-08-08T07:50:56Z

193.190.204.60: /* History */

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[[File:Death-valley-sar.jpg|thumb|right|upright|[[Synthetic aperture radar]] image of [[Death Valley]] colored using polarimetry.]]

'''Polarimetry''' is the measurement and interpretation of the [[Polarization (waves)|polarization]] of [[transverse wave]]s, most notably [[electromagnetic wave]]s, such as radio or light waves. Typically polarimetry is done on electromagnetic waves that have traveled through or have been [[Reflection (physics)|reflected]], [[refracted]], or [[diffracted]] by some material in order to characterize that object.<ref>{{cite book |title=Polarimetric Detection, Characterization and Remote Sensing, Proceedings of the NATO Advanced Study Institute on Special Detection Technique (Polarimetry) and Remote Sensing Yalta, Ukraine 20 September - 1 October 2010, Series: NATO Science for Peace and Security Series C: Environmental Security |editors=Mishchenko, M.I.; Yatskiv, Y.S.; Rosenbush, V.K.; Videen, G. (Eds.) |edition=1st Edition. |year=2011 |url=http://www.springer.com/earth+sciences+and+geography/remote+sensing/book/978-94-007-1635-3 }}</ref><ref>{{cite book |title=Astronomical Polarimetry |author= Jaap Tinbergen Jaap Tinbergen |publisher= Cambridge University Press |isbn= 0-521-01858-7 |year=2007 }}</ref>

==Applications==

Polarimetry of thin films and surfaces is commonly known as [[ellipsometry]].

Polarimetry is used in [[remote sensing]] applications, such as [[planetary science]] and [[weather radar]].

Polarimetry can also be included in computational analysis of waves. For example, radars often consider wave polarization in post-processing to improve the characterization of the targets. In this case, polarimetry can be used to estimate the fine texture of a material, help resolve the orientation of small structures in the target, and, when circularly-polarized antennas are used, resolve the number of bounces of the received signal (the [[chirality (chemistry)|chirality]] of circularly polarized waves alternates with each reflection).

==Equipment==
A [[polarimeter]] is the basic [[Measuring instrument|scientific instrument]] used to make these measurements, although this term is rarely used to describe a polarimetry process performed by a computer, such as is done in polarimetric [[synthetic aperture radar]].

Polarimetry can be used to measure various optical properties of a material, including linear [[birefringence]], circular birefringence (also known as [[optical rotation]] or optical rotary dispersion), [[Dichroism|linear dichroism]], [[circular dichroism]] and [[scattering]].<ref>{{cite book |title=Tissue Optics Light Scattering Methods and Instruments for Medical Diagnosis |author=V. Tuchin |isbn=0-8194-3459-0 |publisher=Society of Photo Optical |year=2000}}</ref> To measure these various properties, there have been many designs of polarimeters. Some are archaic and some are in current use. The most sensitive polarimeters are based on [[interferometer]]s, while more conventional polarimeters are based on arrangements of [[Polarizer|polarising filters]], [[wave plate]]s or other devices.

=== Astronomical polarimetry ===

Light given off by a star is un-polarized, i.e. the direction of oscillation of the light wave is random. However, when the light is reflected off the atmosphere of a planet, the light waves interact with the molecules in the atmosphere and they are polarized.<ref>{{cite journal | author=Schmid, H. M.; Beuzit, J.-L.; Feldt, M. et al. | title=Search and investigation of extra-solar planets with polarimetry | journal=Direct Imaging of Exoplanets: Science & Techniques. Proceedings of the IAU Colloquium #200 | year=2006 | volume= 1| issue= C200| pages=165–170 | bibcode=2006dies.conf..165S | doi=10.1017/S1743921306009252
}}</ref>

By analyzing the polarization in the combined light of an extrasolar planet and its star (about one part in a million), these measurements can in principle be made with very high sensitivity also on ground-based observatories, as polarimetry is not limited by the stability of the Earth's atmosphere. It is akin to [[astronomical transit|transiting]] of a planet in front of its star.

== Measuring optical rotation ==
[[Optical activity|Optically active]] samples, such as solutions of chiral molecules, often exhibit circular [[birefringence]]. Circular birefringence causes rotation of the polarization of plane polarized light as it passes through the sample.

In an Ordinary light, the vibrations occur in all planes perpendicular to direction of propagation. When it is allowed to pass through a [[Nicol prism]] then its vibrations in all directions except the direction of axis of the prism are cut off. The light emerging out of the prism is said to be [[plane polarised]] because its vibration is in one direction. If two Nicol prisms are placed with their polarization planes parallel to each other, then the light rays emerging out of the first prism will enter the second prism. As a result complete bright light is observed. If the second prism is rotated by an angle of 90°, the light emerging from the first prism is stopped by the second prism due to which complete dark or no light region is observed. The first prism is usually called [[polarizer]] and the second prism is called [[analyser]].

A simple polarimeter to measure this rotation consists of a long tube with flat [[glass]] ends, into which the sample is placed. At each end of the tube is a [[Nicol prism]] or other polarizer. [[Visible light|Light]] is shone through the tube, and the prism at the other end, attached to an eye-piece, is rotated to measure the region of complete brightness or half-dark half bright region or complete dark region. The angle of rotation is then read from a scale. The same phenomenon is observed after an angle of 180°. The [[specific rotation]] of the sample may then be calculated. Temperature can affect the rotation of light, which should be accounted for in the calculations.

<math> [\alpha]_\lambda^T = 100\alpha/l\rho\,\!</math>

where:

* [α]λT is the specific rotation.
* T is the temperature.
* λ is the wavelength of light.
* α is the angle of rotation.
* l is the length of the [[polarimeter tube]].
* <math>\rho</math> is the [[mass concentration (chemistry)|mass concentration]] of solution.

==References==
{{reflist|2}}

==External links==
* [http://www.gemstonebuzz.com/instruments/polariscope.html Polariscope - Gemstone Buzz] instrument to measure optical properties.
* [[EU]] Project [[Nanocharm|NanoCharM]] [http://www.nanocharm.org]
{{Use dmy dates|date=December 2010}}
[http://www.testing-instruments.com/pet-preform-instruments/polariscope-computerised-4 Polariscope Computerised ]

{{Exoplanet}}

[[Category:Polarization (waves)]]
[[Category:Optical devices]]