Seven-dimensional cross product

In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group.

The theorem for locally compact abelian groups

Bochner's theorem for a locally compact Abelian group G, with dual group $\hat{G}$, says the following:

Theorem For any normalized continuous positive definite function f on G (normalization here means f is 1 at the unit of G), there exists a unique probability measure on $\hat{G}$ such that

$f(g)=\underset{\hat{G}}{\int }\xi (g)d\mu (\xi ),$

i.e. f is the Fourier transform of a unique probability measure μ on $\hat{G}$. Conversely, the Fourier transform of a probability measure $\hat{G}$ is necessarily a normalized continuous positive definite function f on G. This is in fact a one-to-one correspondence.

The Gelfand-Fourier transform is an isomorphism between the group C*-algebra C*(G) and C₀(G^). The theorem is essentially the dual statement for states of the two Abelian C*-algebras.

The proof of the theorem passes through vector states on strongly continuous unitary representations of G (the proof in fact shows every normalized continuous positive definite function must be of this form).

Given a normalized continuous positive definite function f on G, one can construct a strongly continuous unitary representation of G in a natural way: Let F₀(G) be the family of complex valued functions on G with finite support, i.e. h(g) = 0 for all but finitely many g. The positive definite kernel K(g₁, g₂) = f(g₁ - g₂) induces a (possibly degenerate) inner product on F₀(G). Quotiening out degeneracy and taking the completion gives a Hilbert space

$(\mathcal{\mathscr{H}},⟨,{⟩}_f)$

whose typical element is an equivalence class [h]. For a fixed g in G, the "shift operator" U_g defined by (U_g)( h ) (g') = h(g' - g), for a representative of [h], is unitary. So the map

$g\mapsto U_g$

is a unitary representations of G on $(\mathcal{\mathscr{H}},⟨,{⟩}_f)$. By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have

$⟨U_g[e],[e]{⟩}_f=f(g)$

where [e] is the class of the function that is 1 on the identity of G and zero elsewhere. But by Gelfand-Fourier isomorphism, the vector state $⟨\cdot [e],[e]{⟩}_f$ on C*(G) is the pull-back of a state on $C_0(\hat{G})$, which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives

$⟨U_g[e],[e]{⟩}_f=\underset{\hat{G}}{\int }\xi (g)d\mu (\xi ).$

On the other hand, given a probability measure μ on $\hat{G}$, the function

$f(g)=\underset{\hat{G}}{\int }\xi (g)d\mu (\xi ).$

is a normalized continuous positive definite function. Continuity of f follows from the dominated convergence theorem. For positive definiteness, take a nondegenerate representation of $C_0(\hat{G})$. This extends uniquely to a representation of its multiplier algebra $C_b(\hat{G})$ and therefore a strongly continuous unitary representation U_g. As above we have f given by some vector state on U_g

$f(g)=⟨U_{g}v,v⟩,$

therefore positive-definite.

The two constructions are mutual inverses.

Special cases

Bochner's theorem in the special case of the discrete group Z is often referred to as Herglotz's theorem, (see Herglotz representation theorem) and says that a function f on Z with f(0) = 1 is positive definite if and only if there exists a probability measure μ on the circle T such that

$f(k)=\underset{\mathbb{𝕋}}{\int }{e}^{-2\pi ikx}d\mu (x).$

Similarly, a continuous function f on R with f(0) = 1 is positive definite if and only if there exists a probability measure μ on R such that

$f(t)=\underset{\mathbb{ℝ}}{\int }{e}^{-2\pi i\xi t}d\mu (\xi ).$

Applications

In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series. A sequence of random variables $\{f_n\}$ of mean 0 is a (wide-sense) stationary time series if the covariance

$\text{Cov}(f_n,f_m)$

only depends on n-m. The function

$g(n-m)=\text{Cov}(f_n,f_m)$

is called the autocovariance function of the time series. By the mean zero assumption,

$g(n-m)=⟨f_n,f_m⟩$

where ⟨⋅ , ⋅⟩ denotes the inner product on the Hilbert space of random variables with finite second moments. It is then immediate that g is a positive definite function on the integers ℤ. By Bochner's theorem, there exists a unique positive measure μ on [0, 1] such that

$g(k)=\int e^{-2\pi ikx}d\mu (x)$.

This measure μ is called the spectral measure of the time series. It yields information about the "seasonal trends" of the series.

For example, let z be an m-th root of unity (with the current identification, this is 1/m ∈ [0,1]) and f be a random variable of mean 0 and variance 1. Consider the time series $\{z^{n}f\}$. The autocovariance function is

$g(k)=z^k$.

Evidently the corresponding spectral measure is the Dirac point mass centered at z. This is related to the fact that the time series repeats itself every m periods.

When g has sufficiently fast decay, the measure μ is absolutely continuous with respect to the Lebesgue measure and its Radon-Nikodym derivative f is called the spectral density of the time series. When g lies in l¹(ℤ), f is the Fourier transform of g.

See also

• Positive definite function on a group
• Characteristic function (probability theory)

References

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• M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010