Poisson summation formula

The Poisson summation formula is a tool of Fourier analysis and signal processing. Furthermore, it serves to analyze the properties of sampling methods.

Statement

Be a Schwartz function and is

The continuous Fourier transform of at. Then says the Poisson summation formula

This identity also applies to certain general classes of functions. Suitable conditions are, for example, that the function twice continuously differentiable and the expression is f restricted.

Taking advantage of the basic properties of the Fourier transform, this results in the more general formula with additional parameters,

If, in the more general form,

So the Poisson summation formula can also be read as an identity of a Fourier series of function values ​​of f as coefficients on the left side and a periodization of the Fourier transform of f on the right side. This identity, with the exception of a set of measure zero, if f is a band- limited function, ie the Fourier transform is a measurable function with compact support.

Formulation by Dirac comb

The Dirac comb to the interval length is the distribution

The Fourier transform of a temperature distribution is defined by

In analogy to the Plancherel identity. Since the Fourier transform is a continuous operator on the Schwartz space, this expression actually defines a tempered distribution.

The Dirac comb is a tempered distribution, and the Poisson summation formula states that,

Is. This can also be in the form

. Write The exponential functions are to be considered as tempered distributions, and the series converges in the sense of distributions, ie, in the weak -* sense to the Dirac comb. Note, however, that it converges nowhere in the ordinary sense.

For the proof

Let f be sufficiently smooth and sufficiently rapidly decreasing at infinity, so that the periodization

Ever, limited, differentiable and periodic with period 1. This can therefore be expanded in a pointwise convergent Fourier series,

Their Fourier coefficients are determined by the formula

Also follows from the rapid decay at infinity that the sum with the integral can be interchanged. Therefore, applies with s = t n further

Summarize

Which results in the assertion.

Application to band- limited functions

Let x the band limited with the highest frequency W, that is. If so then enters the right side of the empirical formula only one summand on, with the substitutions t = 0 and receives a multiplication factor to

When multiplied by the indicator function of the interval [- W, W ], and subsequently the inverse Fourier transformation results

In the limiting case, this is the reconstruction formula of Nyquist-Shannon sampling theorem

Where the sinc function is.

Applications in number theory

Using the Poisson's summation formula, one can show that the function of theta

The transformation formula

Sufficient. This transformation formula was used by Bernhard Riemann in the proof of the functional equation of the Riemann zeta function.