Lattice point identities and Shannon-type sampling - Freeden, W. (Willi); Nashed, M. Zuhair (London : CRC Press LLC, 2019)
This book leads the reader through a research excursion, beginning from the Gaussian circle problem of the early nineteenth century, via the classical Hardy-Landau lattice point identity and the Hardy conjecture of the first half of the twentieth century, and the Shannon sampling theorem (its variants, generalizations and the fascinating stories about the cardinal series) of the second half of the twentieth century. The authors demonstrate how all these facets have resulted in new multivariate extensions of lattice point identities and Shannon-type sampling procedures of high practical applicability, thereby also providing a general reproducing kernel Hilbert space structure of an associated Paley-Wiener theory over (potato-like) bounded regions (cf. the cover illustration of the geoid), as well as the whole Euclidean space.
8.6 Hankel Transform and Discontinuous IntegralsPart IV: Multivariate Euler-Type Summation Formulas over Regular Regions; 9. Euler-Green Function and Euler-Type Summation Formula; 9.1 Euler-Green Function; 9.2 Euler-Type Summation Formulas over Regular Regions Based on Euler-Green Functions; 9.3 Iterated Euler-Green Function; 9.4 Euler-Type Summation Formulas over Regular Regions Based on Iterated Euler-Green Functions; Part V: Bivariate Lattice Point/Ball Summation and Shannon-Type Sampling; 10. Hardy-Landau-Type Lattice Point Identities (Constant Weight)
Description based upon print version of record.
10.1 Integral Mean Asymptotics for the Euler-Green Function
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Keywords: Sampling (Statistics); Lattice theory