Heritage and Advancements in Lattice Point and Sampling Regularization Theory
Volume 1: From Gauss' Circle Problem to Lattice Point Distributions
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Beschreibung
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.Establishes a formal analytic connection between lattice point theory and sampling methods via harmonic frameworks Introduces a rigorous multiscale geomathematical framework advancing analytic approaches to lattice point theory Provides a comprehensive operator‑theoretic development of Hardy–Landau theory in fully multivariate settings
Autor*in
Willi Freeden
Themen in »Heritage and Advancements in Lattice Point and Sampling Regularization Theory«
Lattice (Green) functions Euler and Poisson summation formulas Laplace and Helmholtz operators Gauss-Weierstrass integral transform Alternating series Convergence criteria Hardy-Landau lattice point theory Lattice point discrepancy and error Gauss’ circle and sphere problem Multiscale decorrelation and Gauss summability Planar lattice point distribution