Heritage and Advancements in Lattice Point and Sampling Regularization Theory

Volume 2: From Gauss' Integral Transform to Sampling Regularization
Erscheinungsjahr 2026
ISBN: 978-3-032-34658-2
Verlag: Birkhäuser

Buch, Englisch, Format (B × H): 155 mm x 235 mm

Reihe: Geosystems Mathematics

This is the second book of two volumes dedicated to C. F. Gauss on the occasion of his 250th birthday. The objective of the two books is to demonstrate that the heritage of Gauss still has much to offer today in building a strong scientific bridge between mathematics and geoscience.

Volume 2 demonstrates that multidimensional alternating series become more tractable through convergence criteria relating oscillatory properties to an appropriate choice of a Helmholtz operator. As a consequence, new classes of lattice point identities within Shannon sampling can be developed. Wavelet sampling framework is applied to inverse antenna theory. A distinctive feature of sampling is that the coefficients governing the reproduction process for generalized measurements can be determined explicitly using methods originating from lattice point theory. Thus, Volume 2 can be regarded as a forward-looking compendium for addressing ill-posed inverse problems through sampling-based regularization.

This book will be valuable to a broad audience, including mathematicians working in number theory and applied mathematics, as well as professionals in the geosciences, such as geophysics and geoengineering.

Zielgruppe

Research

Autoren/Hrsg.

Freeden, Willi

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Mathematische Analysis Integralrechnungen- und -gleichungen

Inhaltsverzeichnis

Part I. Introduction.- Chapter 1. Objectives and Layout.- Chapter 2. Recovery Concept.- Part II. Univariate Background.- Chapter 3. Integral Transforms.- Chapter 4. Lattice Point Summation and Shannon Sampling.- Chapter 5. Finite Interval Antenna Problem for Arbitrarily Located Data Sets.- Chapter 6. Finite Interval Antenna Problem for Uniformly Located
Data Sets.- Part III. Multivariate Tools.- Chapter 7. Basic Notation and Concepts.- Chapter 8. Metaharmonics.- Chapter 9. Integral Transforms.- Chapter 10. Lattices and Periodization.- Part IV. Summation Formulas over Regular Regions.- Chapter 11. Lattice Point Euler Summation Formulas.- Chapter 12. Lattice Point Poisson Summation Formulas.- Chapter 13. Poisson Lattice Ball Summation Formulas.- Part V. Summation Formulas over Euclidean Spaces.- Chapter 14. Poisson Lattice Point Summation Formulas.- Part VI. Bandlimited and Non-Bandlimited Shannon Sampling.- Chapter 15. Bandlimited Sampling.- Chapter 16. Bandlimited Wavelet Sampling.- Chapter 17. Non-Bandlimited Sampling.- Chapter 18. Non-Bandlimited Wavelet Sampling.- Part VII. Antenna Problem.- Chapter 19. Inverse Ill-Posed Problems in Recovery Context.- Chapter 20. Generalized Measurement Methodologies and Arbitrarily Located Data Systems.- Chapter 21. Uniform Data, Cardinal Series, and Mollifier Inversion.- Chapter 22. Generalized Measurement Methodologies and Uniformly Located Data Systems.- Chapter 23. Paley-Wiener RKHS Methodologies.- Chapter 24. Antenna Context for Euclidean Spaces.- Part VIII. Conclusion.- Chapter 25. Perspectives of Antenna Context in Geoexploration.

Über Autor(innen)

Willi Freeden completed his studies in mathematics, geography, and philosophy at the RWTH Aachen University. In 1971, he obtained his Diploma in Mathematics, in 1972, his Staatsexamen in mathematics and geography, in 1975, his PhD in mathematics, and in 1979, his Habilitation. In 1981/1982, he was a visiting research professor at the Ohio State University, Columbus (Department of Geodetic Science and Surveying). He became a professor of mathematics at the RWTH Aachen University (Institute of Pure and Applied Mathematics) in 1984 and later in 1989, joined the University of Kaiserslautern as a professor of technomathematics (Industrial Mathematics). He was appointed and has been the Head of the Geomathematics Group since 1994 and he was the Vice President for Research and Technology at the University of Kaiserslautern, a position he held from 2002 to 2006.