Metaharmonic Lattice Point Theory Chapman & Hall Pure and Applied Mathematics Series

Metaharmonic Lattice Point Theory covers interrelated methods and tools of spherically oriented geomathematics and periodically reflected analytic number theory. The book establishes multi-dimensional Euler and Poisson summation formulas corresponding to elliptic operators for the adaptive determination and calculation of formulas and identities of weighted lattice point numbers, in particular the non-uniform distribution of lattice points.

The author explains how to obtain multi-dimensional generalizations of the Euler summation formula by interpreting classical Bernoulli polynomials as Green?s functions and linking them to Zeta and Theta functions. To generate multi-dimensional Euler summation formulas on arbitrary lattices, the Helmholtz wave equation must be converted into an associated integral equation using Green?s functions as bridging tools. After doing this, the weighted sums of functional values for a prescribed system of lattice points can be compared with the corresponding integral over the function.

Exploring special function systems of Laplace and Helmholtz equations, this book focuses on the analytic theory of numbers in Euclidean spaces based on methods and procedures of mathematical physics. It shows how these fundamental techniques are used in geomathematical research areas, including gravitation, magnetics, and geothermal.

Introduction. Basic Notation. One-Dimensional Auxiliary Material. One-Dimensional Euler and Poisson Summation Formulas. Preparatory Tools of Analytic Theory of Numbers. Preparatory Tools of Mathematical Physics. Preparatory Tools of Fourier Analysis. Lattice Function for the Iterated Helmholtz Operator. Euler Summation on Regular Regions. Lattice Point Summation. Lattice Ball Summation. Poisson Summation on Regular Regions. Poisson Summation on Planar Regular Regions. Planar Distribution of Lattice Points. Conclusions. Bibliography. Index.

Willi Freeden is the head of the Geomathematics Group in the Department of Mathematics at the University of Kaiserslautern, where he has been vice president for research and technology. Dr. Freeden is also editor-in-chief of the International Journal on Geomathematics. His research interests include special functions of mathematical geophysics, partial differential equations, constructive approximation, numerical methods and scientific computing, and inverse problems in geophysics, geodesy, and satellite technology.