Lattice Point Identities and Shannon-Type Sampling demonstrates that significant roots of many recent&#12288;facets&#12288;of Shannon's sampling theorem for&#12288;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,&#12288;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: <ul> <li>New convergence criteria for alternating series in multi-dimensional analysis</li> <li>Self-contained development of lattice point identities of analytic number theory</li> <li>Innovative lattice point approach to Shannon sampling theory</li> <li>Useful for students of multivariate constructive approximation, and indeed anyone interested in the applicability of signal processing to inverse problems.</li> </ul>

Lattice Point Identities and Shannon-Type Sampling (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

Science & Math

Freeden, Willi

Nashed, M. Zuhair

Mathematics

Chapman and Hall/CRC

Spherical Sampling (Geosystems Mathematics)

Schreiner, Michael

Earth Sciences

Birkhauser

This book presents, in a consistent and unified overview, results and developments in the field of today´s spherical sampling, particularly arising in mathematical geosciences. Although the book often refers to original contributions, the authors made them accessible to (graduate) students and scientists not only from mathematics but also from geosciences and geoengineering. Building a library of topics in spherical sampling theory it shows how advances in this theory lead to new discoveries in mathematical, geodetic, geophysical as well as other scientific branches like neuro-medicine. A must-to-read for everybody working in the area of spherical sampling.

Computers & Technology

Computer Science

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