Paley–Wiener theorem

Paley–Wiener theorem

In mathematics, the Paley–Wiener theorem relates growth properties of entire functions on Cn and Fourier transformation of Schwartz distributions of compact support.

Generally, the Fourier transform can be defined for any tempered distribution; moreover, any distribution of compact support "v" is a tempered distribution. If "v" is a distribution of compact support and "f" is an infinitely differentiable function, the expression

: v(f) = v_x left(f(x) ight)

is well defined. In the above expression the variable "x" in "vx" is a dummy variable and indicates that the distribution is to be applied with the argument function considered as a function of "x".

It can be shown that the Fourier transform of "v" is a function (as opposed to a general tempered distribution) given at the value "s" by

: hat{v}(s) = (2 pi)^{-n/2} v_xleft(e^{-i langle x, s angle} ight)

and that this function can be extended to values of "s" in the complex space Cn. This extension of the Fourier transform to the complex domain is called the Fourier-Laplace transform.

Theorem. An entire function "F" on Cn is the Fourier-Laplace transform of distribution "v" of compact support if and only if for all "z" ∈ C"n",

: |F(z)| leq C (1 + |z|)^N e^{B| mathfrak{Im} z

for some constants "C", "N", "B". The distribution "v" in fact will be supported in the closed ball of center 0 and radius "B".

Additional growth conditions on the entire function "F" impose regularity properties on the distribution "v": For instance, if for "every" positive "N" there is a constant "CN" such that for all "z" ∈ C"n",

: |F(z)| leq C_N (1 + |z|)^{-N} e^{B| mathfrak{Im} z

then "v" is infinitely differentiable and conversely.

The theorem is named for Raymond Paley (1907 - 1933) and Norbert Wiener (1894 - 1964). Their formulations were not in terms of distributions, a concept not at the time available. The formulation presented here is attributed to Lars Hörmander.

In another version, the Paley–Wiener theorem explicitly describes the Hardy space H^2(mathbf{R}) using the unitary Fourier transform mathcal{F}. The theorem states that: mathcal{F}H^2(mathbf{R})=L^2(mathbf{R_+}).This is a very useful result as it enables one pass to the Fourier transform of a function in the Hardy space and perform calculations in the easily understood space L^2(mathbf{R_+}) of square-integrable functions supported on the positive axis.

References

See section 3 Chapter VI of

* K. Yosida, "Functional Analysis", Academic Press, 1968

See also Theorem 1.7.7 in

* L. Hörmander, "Linear Partial Differential Operators", Springer Verlag, 1976

See Paley–Wiener Theorems (7.22 - 7.23) of:

* W. Rudin, "Functional Analysis", McGraw-Hill Book Company, 1973 First Edition


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Paley-Wiener integral — In mathematics, the Paley Wiener integral is a simple stochastic integral. When applied to classical Wiener space, it is less general than the Itō integral, but the two agree when they are both defined.The integral is named after its discoverers …   Wikipedia

  • Théorème de Paley-Wiener — En mathématiques, on appelle théorème de Paley Wiener tout théorème qui relie les propriétés de décroissance à l infini d une fonction ou d une distribution avec l analyticité de sa transformée de Fourier. Ce théorème a été ainsi nommé en hommage …   Wikipédia en Français

  • Wiener — is German for Viennese, but may also refer to: * A hot dog, from German Wiener Würstchen , meaning Viennese small sausage * A slang term for penis Wiener is the surname of: * Alexander S. Wiener (1907 76), leader in the fields of forensic… …   Wikipedia

  • Raymond Paley — Raymond Edward Alan Christopher Paley (7 January 1907 ndash; 7 April 1933) was an English mathematician. Paley was born in Bournemouth, England. He was educated at Eton. From there he entered Trinity College, Cambridge where he showed himself the …   Wikipedia

  • Norbert Wiener — Born November 26, 1894(1894 11 26) Columbia, Missouri, U.S …   Wikipedia

  • Norbert Wiener — (* 26. November 1894 in Columbia, Missouri; † 18. März 1964 in Stockholm) war ein US amerikanischer Mathematiker. Er ist als Begründer der Kybernetik bekannt, ein Ausdruck, den er in seinem Werk Cybernetics or Control and Communication in the… …   Deutsch Wikipedia

  • Cameron-Martin theorem — In mathematics, the Cameron Martin theorem or Cameron Martin formula is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron Martin Hilbert space.MotivationRecall that …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • Norbert Weiner — Norbert Wiener (* 26. November 1894 in Columbia, Missouri; † 18. März 1964 in Stockholm) war ein amerikanischer Mathematiker. Er ist als Begründer der Kybernetik bekannt, ein Ausdruck, den er in seinem Werk Cybernetics or Control and… …   Deutsch Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”