From The Gaussian Circle Problem To Multivariate Shannon Sampling

Keywords

Explicit over- and undersampling involving regular regions; Gaussian circle problem; Gaussian summability of cardinal series; Hardy-type lattice point summation; Paley-Wiener spaces; Shannon-type sampling; Sinc-type reproducing kernels

Abstract

A historical overview leading to present generalizations of Shannon's sampling theorem (1949) and starting from the Gaussian circle problem (1801) is sketched. It is shown that the bridge between Gauss's and Shannon's work is constituted by certain extensions of the famous Hardy-Landau identities in geometric lattice point theory. Particular interest is laid on the matter dealing with bandlimited functions corresponding to, e.g., geoscientifically relevant regions. Emphasis is put on the study of the convergence of the resulting multivariate cardinal series, as well as on the delicate explicit specification of under- and oversampling in multivariate Shannon sampling. Finally, the routes to sampling expansions are exhibited in Paley-Wiener spaces, leading to multivariate sinc-type reproducing kernels.

Publication Date

1-12-2018

Publication Title

Frontiers In Orthogonal Polynomials and Q-series

Volume

1

Number of Pages

213-237

Document Type

Article; Book Chapter

Personal Identifier

scopus

DOI Link

https://doi.org/10.1142/9789813228887_0011

Socpus ID

85045713787 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85045713787

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