Heritage and Advancements in Lattice Point and Sampling Regularization Theory

Volume 1: From Gauss' Circle Problem to Lattice Point Distributions
Erscheinungsjahr 2026
ISBN: 978-3-032-34662-9
Verlag: Birkhäuser

Buch, Englisch, 457 Seiten, Format (B × H): 155 mm x 235 mm

Reihe: Geosystems Mathematics

This is the first 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 1 presents a proof of Hardy’s conjecture for the Gauss circle problem in geometric number theory. In fact, the verification of this conjecture represents a consequence of the “transfer” of methods and frameworks from geomathematics to number theory. The essential ingredients originate from tools in mathematical (geo-)physics. Remarkably, the foundational techniques enable the recovery of significant topics in lattice point theory, including the Hardy–Landau identities and the planar non-uniform distribution of lattice points. Thus, Volume 1 can be characterized as a contribution to the geometric theory of numbers, strongly influenced by methods and procedures originating in geosystems mathematics.   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 Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Integralrechnungen- und -gleichungen

Inhaltsverzeichnis

Part I. Introduction.- Chapter 1. Key Idea and Objective.- Chapter 2. Goal, Layout, and Roots.- Part II. Univariate Background.- Chapter 3. Integral Transforms.- Chapter 4. Summation Formulas.- Part III. Multivariate Tools.- Chapter 5. Nomenclature and Basic Concepts.- Chapter 6. Spherical Harmonics and Bessel Functions.- Chapter 7. Integral Transforms.- Chapter 8. Lattices.- Chapter 9. Lattice Points inside Spheres.- Chapter 10. Lattice Points on Spheres.- Chapter 11. Periodicity and Periodization.- Part IV. Summation Formulas over Regular Regions with Respect to Helmholtz Operators.- Chapter 12. Lattice Point Euler Summation Formulas.- Chapter 13. Lattice Function and Its Representations.- Chapter 14. Lattice Point Poisson Summation Formulas.- Chapter 15. Lattice Ball Poisson Summation Formulas.- Part V. Summation Formulas over Euclidean Spaces with Respect to Helmholtz Operators.- Chapter 16. Lattice Point Poisson Summation Formulas.- Part VI. Hardy-Landau Theory.- Chapter 17. Hardy-Landau Lattice Point Theory.- Chapter 18. Weighted Hardy-Landau Lattice Point Theory.- Part VII. Solvability of the Circle Problem.- Chapter 19. Previous Findings.- Chapter 20. Hardy’s Conjecture for the Unit Lattice.- Chapter 21. Hardy’s Conjecture for Arbitrary Lattices.- Chapter 22. Angular Weight Extension of the Circle Problem.- Chapter 23. Higher Dimensional Extensions of the Circle Problem.- Part VIII. Further Aspects.- Chapter 24. Non-Uniform Distribution of Lattice Points.

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.