ABSTRACT

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.

Features:

  • New convergence criteria for alternating series in multi-dimensional analysis
  • Self-contained development of lattice point identities of analytic number theory
  • Innovative lattice point approach to Shannon sampling theory
  • Useful for students of multivariate constructive approximation, and indeed anyone interested in the applicability of signal processing to inverse problems.

part Part 1|1 pages

Central Theme

chapter Chapter 3|5 pages

Layout

part Part II|1 pages

Univariate Poisson-Type Summation Formulas and Shannon-Type Sampling

part Part III|1 pages

Preparatory Material for Multivariate Lattice Point Summation and Shannon-Type Sampling

chapter Chapter 5|10 pages

Preparatory Tools of Vector Analysis

chapter Chapter 6|22 pages

Preparatory Tools of the Theory of Special Functions

chapter Chapter 7|16 pages

Preparatory Tools of Lattice Point Theory

chapter Chapter 8|17 pages

Preparatory Tools of Fourier Analysis

part Part IV|1 pages

Multivariate Euler-Type Summation Formulas over Regular Regions

part Part V|1 pages

Bivariate Lattice Point/Ball Summation and Shannon-Type Sampling

part Part VI|1 pages

Multivariate Poisson-Type Summation Formulas over Regular Regions

part Part VII|1 pages

Multivariate Shannon-Type Sampling Formulas over Regular Regions

part Part VIII|1 pages

Multivariate Poisson-Type Summation Formulas over Euclidean Spaces

chapter Chapter 18|24 pages

Poisson-Type Summation Formulas over Euclidean Spaces

part Part IX|1 pages

Multivariate Shannon-Type Sampling Formulas over Euclidean Spaces

part Part X|1 pages

Conclusion

chapter Chapter 20|5 pages

Trends, Progress, and Perspectives