Lattice Point Identities and Shannon-Type Sampling Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series
Auteurs : Freeden Willi, Nashed M. Zuhair
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.
Date de parution : 10-2019
15.6x23.4 cm
Thèmes de Lattice Point Identities and Shannon-Type Sampling :
Mots-clés :
Reproducing Kernel Hilbert Space Theory; multivariate constructive approximation; Euler Summation Formula; signal processing; Weierstrass Transform; Shannon-type sampling; Regular Region; Auxiliary Function; Summation Formula; Hardy Landau Identities; Beltrami Operator; Homogeneous Harmonic Polynomial; Poisson Summation Formula; Arbitrary Lattices; Fourier Inversion Formula; Reproducing Kernel Hilbert Space; Lattice Point Theory; Paley Wiener Spaces; Cardinal Series; Lattice Point Sums; Hankel Transform; Shannon Sampling; Reproducing Kernel; Periodic Polynomials; Helmholtz Operator; Inverse Lattice; Analytic Number Theory; Aliasing Error