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← Older revision		Revision as of 01:06, 9 October 2013
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	~~Wilber Berryhill~~ is ~~what his wife loves~~ to ~~contact him~~ and ~~he totally loves this name~~. ~~Ohio~~ is ~~where my home~~ is ~~but my husband wants us~~ to ~~move~~. ~~To climb~~ is ~~something I really appreciate doing~~. ~~She works~~ as a ~~travel agent~~ but ~~quickly she'll~~ be on ~~her personal~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my web blog~~ - [~~http:~~//~~www~~.~~weddingwall~~.~~com~~.au/~~groups~~/~~easy~~-~~advice~~-~~for~~-~~successful~~-~~personal~~-~~development-today/ accurate psychic predictions~~]		In [[mathematical analysis]] an '''oscillatory integral''' is a type of [[distribution (mathematics)\|distribution]]. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals.

			==Definition==

			An oscillatory integral <math> f(x) </math> is written formally as

			:<math> f(x) = \int e^{i \phi(x,\xi)}\, a(x,\xi) \, \mathrm{d} \xi </math>

			where <math> \phi(x,\xi) </math> and <math> a(x,\xi) </math> are functions defined on <math> \mathbb{R}_x^n \times \mathrm{R}^N_\xi </math> with the following properties.

			:1) The function <math> \phi </math> is real valued, [[homogeneous function#Positive homogeneity\|positive homogeneous]] of degree 1, and infinitely differentiable away from <math> \{\xi = 0 \} </math>. Also, we assume that <math> \phi </math> does not have any [[critical point (mathematics)#Several variables\|critical points]] on the [[support (mathematics)\|support]] of <math> a </math>. Such a function, <math> \phi </math> is usually called a '''phase function'''. In some contexts more general functions are considered, and still referred to as phase functions.

			:2) The function <math> a </math> belongs to one of the [[Pseudo-differential_operator#Symbol_Classes_and_Pseudo-Differential_Operators\|symbol classes]] <math> S^m_{1,0}(\mathbb{R}_x^n \times \mathrm{R}^N_\xi) </math> for some <math> m \in \mathbb{R}</math>. Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree <math> m </math>. As with the phase function <math> \phi </math>, in some cases the function <math> a </math> is taken to be in more general, or just different, classes.

			When <math> m < -n + 1 </math> the formal integral defining <math> f(x) </math> converges for all <math> x </math> and there is no need for any further discussion of the definition of <math> f(x) </math>. However, when <math> m \geq - n+1</math> the oscillatory integral is still defined as a distribution on <math> \mathbb{R}^n </math> even though the integral may not converge. In this case the distribution <math> f(x) </math> is defined by using the fact that <math> a(x,\xi) \in S^m_{1,0}(\mathbb{R}_x^n \times \mathrm{R}^N_\xi) </math> may be approximated by functions that have exponential decay in <math> \xi</math>. One possible way to do this is by setting

			:<math> f(x) = \lim \limits_{\epsilon \rightarrow 0^+} \int e^{i \phi(x,\xi)}\, a(x,\xi) e^{-\epsilon \|\xi\|^2/2} \, \mathrm{d} \xi </math>

			where the limit is taken in the sense of [[Tempered_distributions#Tempered_distribution\|tempered distributions]]. Using integration by parts it is possible to show that this limit is well defined, and that there exists a [[differential operator]] <math> L </math> such that the resulting distribution <math> f(x) </math> acting on any <math> \psi </math> in the [[Schwartz space]] is given by

			:<math> \langle f, \psi \rangle = \int e^{i \phi(x,\xi)} L \left ( a(x,\xi) \, \psi(x) \right ) \, \mathrm{d} x \, \mathrm{d} \xi </math>

			where this integral converges absolutely. The operator <math> L </math> is not uniquely defined, but can be chosen in such a way that depends only on the phase function <math> \phi </math>, the order <math> m </math> of the symbol <math> a </math>, and <math> N</math>. In fact, given any integer <math> M </math> it is possible to find an operator <math> L </math> so that the integrand above is bounded by <math> C (1 + \|\xi\|)^{-M} </math> for <math> \|\xi\| </math> sufficiently large. This is the main purpose of the definition of the [[Pseudo-differential_operator#Symbol_Classes_and_Pseudo-Differential_Operators\|symbol classes]].

			==Examples==

			Many familiar distributions can be written as oscillatory integrals.

			:1) The [[Fourier inversion theorem]] implies that the [[Dirac delta function\|delta function]], <math> \delta(x) </math> is equal to

			::<math> \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} e^{i x \cdot \xi} \, \mathrm{d} \xi. </math>

			:If we apply the first method of defining this oscillatory integral from above, as well as the Fourier transform of the Gaussian, we obtain a well known sequence of functions which approximate the delta function:

			::<math> \delta(x) = \lim_{\varepsilon \rightarrow 0^+}\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} e^{i x\cdot \xi} e^{-\varepsilon \|\xi\|^2/2} \mathrm{d} \xi = \lim_{\varepsilon \rightarrow 0^+} \frac{1}{(\sqrt{2 \pi \varepsilon})^n} e^{-\|\xi\|^2/(2 \varepsilon)}. </math>

			:An operator <math> L </math> in this case is given for example by

			::<math> L = \frac{(1 - \Delta_x)^k}{(1 + \|\xi\|^2)^k} </math>

			:where <math> \Delta_x </math> is the [[Laplace operator\|Laplacian]] with respect to the <math> x </math> variables, and <math> k </math> is any integer greater than <math> (n-1)/2</math>. Indeed, with this <math> L </math> we have

			::<math> \langle \delta, \psi \rangle = \psi(0) = \frac{1}{(2 \pi)^n}\int_{\mathbb{R}^n} e^{i x \cdot \xi} L(\psi)(x,\xi)\, \mathrm{d} \xi \, \mathrm{d} x, </math>

			:and this integral converges absolutely.

			:2) The [[Schwartz kernel theorem\|Schwartz kernel]] of any differential operator can be written as an oscillatory integral. Indeed if

			::<math> L = \sum \limits_{\|\alpha\| \leq m} p_\alpha(x) D^\alpha </math>

			:where <math> D^\alpha = \partial^\alpha_{x}/i^{\|\alpha\|} </math>, then the kernel of <math> L </math> is given by

			::<math> \frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{i \xi \cdot (x - y)} \sum \limits_{\|\alpha\| \leq m} p_\alpha(x) \, \xi^\alpha \, \mathrm{d} \xi. </math>

			==Relation to Lagrangian distributions==

			Any Lagrangian distribution can be represented locally by oscillatory integrals (see {{harvtxt\|Hörmander\|1983}}). Conversely any oscillatory integral is a Lagrangian distribution. This gives a precise description of the types of distributions which may be represented as oscillatory integrals.

			== See also ==

			* [[Riemann–Lebesgue lemma]]
			* [[van der Corput lemma (harmonic analysis)\|van der Corput lemma]]

			==References==

			*{{Citation\|first=Lars\|last=[[Lars Hörmander\|Hörmander]]\|title=The Analysis of Linear Partial Differential Operators IV\|publisher=Springer-Verlag\|year=1983\|isbn=0-387-13829-3}}

			*{{Citation\|first=Lars\|last=[[Lars Hörmander\|Hörmander]]\|title=Fourier integral operators I\|journal=Acta Math.\|year=1971\|volume=127\|pages=79–183}}

			[[Category:Mathematical analysis]]

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Wilber Berryhill is what his wife loves to contact him and he totally loves this name. Ohio is where my home is but my husband wants us to move. To climb is something I really appreciate doing. She works as a travel agent but quickly she'll be on her personal.<br><br>Feel free to surf to my web blog - [http://www.weddingwall.com.au/groups/easy-advice-for-successful-personal-development-today/ accurate psychic predictions]

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en>Ihaveacatonmydesk: +harv

en>Vatsal4592: Cleaned up using AutoEd; formatting: 2x nbsp-dash (using Advisor.js)

en>Acbyram: Added a citation