Generalized Poisson Summation Formulas For Continuous Functions Of Polynomial Growth

Keywords

Poisson summation formula; Polynomially growing functions; Sampling theory; Tempered distributions; Weighted Sobolev spaces

Abstract

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of L1 signals. This L1 assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that d/ 2 + ε derivatives of the signal are bounded by some polynomial in the L2 sense.

Publication Date

4-1-2017

Publication Title

Journal of Fourier Analysis and Applications

Volume

23

Issue

2

Number of Pages

442-461

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00041-016-9475-9

Socpus ID

84976648467 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84976648467

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