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.

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.