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.
- 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.
Table of Contents
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.
Willi Freeden studied Mathematics, Geography, and Philosophy. He received his Philosophicum 1970, Diplom in Mathematics 1971, Staatsexamen in Mathematics and Geography 1972, Ph.D. in Mathematics 1975, Habilitation in Mathematics 1979, all from RWTH Aachen University, Germany. He has served for many years as a Professor at the RWTH Aachen University and the University of Kaiser-slautern. He has held various visiting professor positions, e.g., at the Ohio State University, Columbus (Department of Geodetic Science and Surveying). He is recipient of the RWTH Borchers Award, the Eurasian Association on Inverse Problems (EAIP) Award, and the Fellowship of the International Association of Geodesy (IAG). Since 1996 he is a member of the German Geodetic Commission of the Bavarian Academy of Sciences, Munich. He had the position of the Vice-President for Research and Technology at the University of Kaiserslautern from 2002 to 2006. He is author, editor, and coeditor of 19 books, published more than 220 papers, several expository papers, and book chapters. He is the (founding) Editor-in-Chief of the Springer “GEM International Journal on Geomathematics”, Editor-in-Chief of the Springer “Handbook of Geomathematics”, Editor-in-Chief of the (German) Springer-Spektrum “Handbuch Tiefe Geothermie” and “Hand-buch Oberﬂ¨achennahe Geothermie”, Editor-in-Chief of the Birkh¨auser book se-ries “Geosystems Mathematics”, Editor-in-Chief of the Birkh¨auser lecture notes “Geosystems Mathematics and Computing”, 2015 Editor-in-Chief of the second edition of the Springer “Handbook of Geomathematics”, 2018 Editor-in-Chief of the Birkh¨auser “Handbook of Mathematical Geodesy”, Editor-in-Chief of the Springer-Spektrum ”Handbuch der Geod¨asie”. He is member of the editorial board of a large number of international journals. He is the organizer of several Oberwol-fach conferences, mini-symposia and Special Sessions at meetings of the American Mathematical Society.
M. Zuhair Nashed received his S.B. and S.M. degrees in Electrical Engineering from MIT and his Ph.D. in Mathematics from the University of Michigan. He has served for many years as a Professor at Georgia Tech and the University of Delaware and has held visiting professor positions at the University of Michigan, University of Wisconsin, AUB, and KFUPM. He has held distinguished visiting scholar positions at many universities around the world. He is a fellow of the Amer-ican Mathematical Society (Inaugural Class of 2013) and the recipient of the Lester Ford Award of the Mathematical Association of America, the Sigma Xi Faculty Research Award and Sustained Research Award in Science from Georgia Tech, Dr. Zakir Husain Award of the Indian Society of Industrial and Applied Mathematics, as well as several other international awards. He has published over 140 papers in mathematics and 30 papers in applied sciences, physics and engineering, has written 30 expository papers and book chapters, and authored and edited 15 books. He is Editor-in-Chief of “Numerical Functional Analysis and Optimization”, Ex-ecutive Editor of “Sampling Theory in Signal and Image Processing”, founding and past coeditor of the “Journal of Integral Equations”, and a member of the ed-itorial board of 36 journals. He is also Editor-in-Chief of the Springer “Handbook of Geomathematics”, Editor-in-Chief of the Birkh¨auser book series “Geosystems Mathematics” and of the Birkh¨auser “Lecture Notes on Geosystems Mathematics and Computing”, and Editor-in-Chief of “Handbook of Mathematical Geodesy”. He has served for 40 years as Executive Editor of the book series “Pure and Ap-plied Mathematics: A Program of Monographs, Textbooks, and Lecture Notes”, published by Marcel Dekker, Inc., and later by Chapman & Hall /CRC. He is Editor-in-Chief of the book series “Contemporary Mathematics and Its Applica-tions”, launched recently by World Scientiﬁc. He was invited to give an hour lecture at the American Mathematical Society and has also given three plenary lectures at meetings of the Mathematical Association of America and plenary lectures at meetings of the French, Tunisian and Lebanese Mathematical Societies, Indian Society of Industrial and Applied Mathematics, and Japan Society of Mechanical Engineers. He gave over 400 plenary and invited talks at conferences and colloquia. He has organized over 40 conferences, mini-symposia, and special sessions.