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Multi-variate Hardy-type lattice point summation and Shannon-type sampling

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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.

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Acknowledgments

The authors thank their colleagues Akram Aldroubi, Mathematics Department, Vanderbilt University, Nashville (USA), Gunter Malle, Mathematics Department, University of Kaiserslautern (Germany), and Gianluca Vinti, Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia (Italy) for valuable comments during the review process.

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  1. Geomathematics Group, University of Kaiserslautern, MPI-Building 26, 67663, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    M. Zuhair Nashed

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  1. Willi Freeden
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  2. M. Zuhair Nashed
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Freeden, W., Nashed, M.Z. Multi-variate Hardy-type lattice point summation and Shannon-type sampling. Int J Geomath 6, 163–249 (2015). https://doi.org/10.1007/s13137-015-0076-6

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