1st Edition

Metaharmonic Lattice Point Theory

By Willi Freeden Copyright 2011
    476 Pages 35 B/W Illustrations
    by Chapman & Hall

    472 Pages 35 B/W Illustrations
    by Chapman & Hall

    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
    Historical Aspects
    Preparatory Ideas and Concepts
    Tasks and Perspectives

    Basic Notation
    Cartesian Nomenclature
    Regular Regions
    Spherical Nomenclature
    Radial and Angular Functions

    One-Dimensional Auxiliary Material
    Gamma Function and Its Properties
    Riemann–Lebesgue Limits
    Fourier Boundary and Stationary Point Asymptotics
    Abel–Poisson and Gauss–Weierstrass Limits

    One-Dimensional Euler and Poisson Summation Formulas
    Lattice Function
    Euler Summation Formula for the Laplace Operator
    Riemann Zeta Function and Lattice Function
    Poisson Summation Formula for the Laplace Operator
    Euler Summation Formula for Helmholtz Operators
    Poisson Summation Formula for Helmholtz Operators

    Preparatory Tools of Analytic Theory of Numbers
    Lattices in Euclidean Spaces
    Basic Results of the Geometry of Numbers
    Lattice Points Inside Circles
    Lattice Points on Circles
    Lattice Points Inside Spheres
    Lattice Points on Spheres

    Preparatory Tools of Mathematical Physics
    Integral Theorems for the Laplace Operator
    Integral Theorems for the Laplace–Beltrami Operator
    Tools Involving the Laplace Operator
    Radial and Angular Decomposition of Harmonics
    Integral Theorems for the Helmholtz–Beltrami Operator
    Radial and Angular Decomposition of Metaharmonics
    Tools Involving Helmholtz Operators

    Preparatory Tools of Fourier Analysis
    Periodical Polynomials and Fourier Expansions
    Classical Fourier Transform
    Poisson Summation and Periodization
    Gauss–Weierstrass and Abel–Poisson Transforms
    Hankel Transform and Discontinuous Integrals

    Lattice Function for the Iterated Helmholtz Operator
    Lattice Function for the Helmholtz Operator
    Lattice Function for the Iterated Helmholtz Operator
    Lattice Function in Terms of Circular Harmonics
    Lattice Function in Terms of Spherical Harmonics

    Euler Summation on Regular Regions
    Euler Summation Formula for the Iterated Laplace Operator
    Lattice Point Discrepancy Involving the Laplace Operator
    Zeta Function and Lattice Function
    Euler Summation Formulas for Iterated Helmholtz Operators
    Lattice Point Discrepancy Involving the Helmholtz Operator

    Lattice Point Summation
    Integral Asymptotics for (Iterated) Lattice Functions
    Convergence Criteria and Theorems
    Lattice Point-Generated Poisson Summation Formula
    Classical Two-Dimensional Hardy–Landau Identity
    Multi-Dimensional Hardy–Landau Identities

    Lattice Ball Summation
    Lattice Ball-Generated Euler Summation Formulas
    Lattice Ball Discrepancy Involving the Laplacian
    Convergence Criteria and Theorems
    Lattice Ball-Generated Poisson Summation Formula
    Multi-Dimensional Hardy–Landau Identities

    Poisson Summation on Regular Regions
    Theta Function and Gauss–Weierstrass Summability
    Convergence Criteria for the Poisson Series
    Generalized Parseval Identity
    Minkowski’s Lattice Point Theorem

    Poisson Summation on Planar Regular Regions
    Fourier Inversion Formula
    Weighted Two-Dimensional Lattice Point Identities
    Weighted Two-Dimensional Lattice Ball Identities

    Planar Distribution of Lattice Points
    Qualitative Hardy–Landau Induced Geometric Interpretation
    Constant Weight Discrepancy
    Almost Periodicity of the Constant Weight Discrepancy
    Angular Weight Discrepancy
    Almost Periodicity of the Angular Weight Discrepancy
    Radial and Angular Weights
    Non-Uniform Distribution of Lattice Points
    Quantitative Step Function Oriented Geometric Interpretation

    Conclusions
    Summary
    Outlook

    Bibliography

    Index

    Biography

    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.