Multi-Variate Hardy-Type Lattice Point Summation And Shannon-Type Sampling

Keywords

Explicit over- and undersampling involving (geoscientifically relevant) regular regions; Gaussian summability of cardinal series; Hardy-type lattice point summation; Parseval-type identities; Shannon-type sampling; Spline interpolation in Paley–Wiener spaces

Abstract

The famous Shannon sampling theorem gives an answer to the question of how a one-dimensional time-dependent bandlimited signal can be reconstructed from discrete values in lattice points. In this work, we are concerned with multi-variate Hardy-type lattice point identities from which space-dependent Shannon-type sampling theorems can be obtained by straightforward integration over certain regular regions. An answer is given to the problem of how a signal bandlimited to a regular region in q-dimensional Euclidean space allows a reconstruction from discrete values in the lattice points of a (general) q-dimensional lattice. Weighted Hardy-type lattice point formulas are derived to allow explicit characterizations of over- and undersampling, thereby specifying not only the occurrence, but also the type of aliasing in a thorough mathematical description. An essential tool for the proof of Hardy-type identities in lattice point theory is the extension of the Euler summation formula to second order Helmholtz-type operators involving associated Green functions with respect to the “boundary condition” of periodicity. In order to circumvent convergence difficulties and/or slow convergence in multi-variate Hardy-type lattice point summation, some summability methods are necessary, namely lattice ball and Gauß–Weierstraß averaging. As a consequence, multi-variate Shannon-type lattice sampling becomes available in a proposed summability context to accelerate the summation of the associated cardinal-type series. Finally, some aspects of constructive approximation in a resulting Paley–Wiener framework are indicated, such as the recovery of a finite set of lost samples, the reproducing Hilbert space context of spline interpolation.

Publication Date

11-1-2015

Publication Title

GEM - International Journal on Geomathematics

Volume

6

Issue

2

Number of Pages

163-249

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s13137-015-0076-6

Socpus ID

84945178949 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84945178949

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