Paley–Wiener theorem

The set of Paley -Wiener, named after Raymond Paley and Norbert Wiener, is a phrase from the mathematical branch of functional analysis. He characterizes the Fourier - Laplace transforms of smooth functions with compact support or tempered distributions with compact support by means of growth conditions.

Introduction

Is a smooth function, so you can know, the Fourier transform

Form, and wherein the dot product of the vectors. This formula is also useful for complex vectors. This is called

The Fourier transform of Laplace. By dualization can extend this conceptualization to distributions with compact support. Is a tempered distribution, then by

The Fourier transform is defined. It should only be noted that a smooth function and that the distributions with compact support are exactly the steady, linear functionals on the space of smooth functions. The above formula can obviously write for and is called

Again the Fourier -Laplace transform of.

The Fourier - Laplace transforms are holomorphic functions and it begs the question as to which holomorphic functions can occur as a Fourier -Laplace transformations here. This is the question answered by the set of Paley -Wiener.

Set of Paley -Wiener functions for

A holomorphic function is exactly what it is the Fourier - Laplace transform of a smooth function with support in the ball, if there is any real constant a, so that

For everyone.

Here, the real vector of the imaginary parts of components of the vector.

Set of Paley -Wiener for distributions

A holomorphic function is exactly what it is the Fourier -Laplace transform of a distribution with support in the ball when and are constants, so that

For everyone.

Remark

The condition in the set of functions is more restrictive than the condition in the theorem for distributions. This is not surprising, since any smooth functions with compact support defined by a distribution with compact support, which is in support of, and applies to the Fourier - Laplace transforms

That is, the Fourier Laplace transform of a smooth function with compact support is the Fourier transform of the Laplacian distribution is defined by them in a compact vehicle.

Examples

The sets of Paley -Wiener will be explained using two examples.

Be first. The Fourier transform is Laplace

If the decomposition into real and imaginary parts, as is, that is, increases as the real part of the solid, at any rate faster than for each constant. This is reflected according to the above rates reflect the fact that no compact support,

Let now the distribution. A short calculation shows

Which is to continue for ever. If the decomposition into real and imaginary parts, the following applies, that is, it can be estimated against, because the hyperbolic functions allow such assessment. It follows that the growth condition from the set of Paley -Wiener for distributions met. In fact, a distribution with compact support. However, the holomorphic function does not satisfy the condition of the theorem of Paley -Wiener functions for, because there would be a constant as in the sentence, followed

Especially for real is the exponential term equal to 1 and there followed, and so would the sine function can be found for large real arguments against 0, but not known to be the case. Although the distribution comes from the characteristic function of the interval [-1,1] ago, and this also has compact support, but it is not smooth.