Lattice Point Identities and Shannon-Type Sampling demonstrates that significant roots of many recent facets of Shannon's sampling theorem for multivariate signals rest on basic number-theoretic results.
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.
All in all, the context of this book represents the fruits of cross-fertilization of various subjects, namely elliptic partial differential equations, Fourier inversion theory, constructive approximation involving Euler and Poisson summation formulas, inverse problems reflecting the multivariate antenna problem, and aspects of analytic and geometric number theory.
Preface. About the Authors. Acknowledgment. 1.From Lattice Point to Shannon-Type Sampling Identities. 2.Obligations, Ingredients, Achievements, and Innovations. 3.Layout. 4.Euler/Poisson-Type Summation Formulas and Shannon-Type Sampling. 5.Preparatory Tools of Vector Analysis. 6.Preparatory Tools of the Theory of Special Functions. 7.Preparatory Tools of Lattice Point Theory. 8.Preparatory Tools of Fourier Analysis. 9.Euler–Green Function and Euler-Type Summation Formula. 10.Hardy–Landau-Type Lattice Point Identities (Constant Weight). 11.Hardy–Landau-Type Lattice Point Identities (General Weights). 12.Bandlimited Shannon-Type Sampling (Preparatory Results). 13.Lattice Ball Shannon-Type Sampling. 14.Gauss-Weierstrass Mean Euler-Type Summation Formulas and Shannon-Type Sampling. 15.From Gauss-Weierstrass to Ordinary Lattice Point Poisson–Type Summation. 16.Shannon-Type Sampling Based on Poisson-Type Summation Formulas. 17.Paley–Wiener Space Framework and Spline Approximation. 18.Poisson-Type Summation Formulas over Euclidean Spaces. 19.Shannon–Type Sampling Based on Poisson–Type Summation Formulas over Euclidean Spaces. 20.Trends, Progress, and Perspectives. Bibliography. Index.