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El. knyga: Metaharmonic Lattice Point Theory

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(Technical University of Kaiserslautern, Germany)
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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 Greens 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 Greens 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.
Preface xi
About the Book xv
About the Author xvii
List of Symbols
xix
List of Figures
xxiii
1 Introduction
1(8)
1.1 Historical Aspects
1(2)
1.2 Preparatory Ideas and Concepts
3(2)
1.3 Tasks and Perspectives
5(4)
2 Basic Notation
9(8)
2.1 Cartesian Nomenclature
9(3)
2.2 Regular Regions
12(1)
2.3 Spherical Nomenclature
13(2)
2.4 Radial and Angular Functions
15(2)
3 One-Dimensional Auxiliary Material
17(28)
3.1 Gamma Function and Its Properties
17(14)
3.2 Riemann-Lebesgue Limits
31(3)
3.3 Fourier Boundary and Stationary Point Asymptotics
34(4)
3.4 Abel-Poisson and Gauß-Weierstraß Limits
38(7)
4 One-Dimensional Euler and Poisson Summation Formulas
45(42)
4.1 Lattice Function
46(8)
4.2 Euler Summation Formula for the Laplace Operator
54(9)
4.3 Riemann Zeta Function and Lattice Function
63(5)
4.4 Poisson Summation Formula for the Laplace Operator
68(8)
4.5 Euler Summation Formula for Helmholtz Operators
76(6)
4.6 Poisson Summation Formula for Helmholtz Operators
82(5)
5 Preparatory Tools of Analytic Theory of Numbers
87(34)
5.1 Lattices in Euclidean Spaces
88(4)
5.2 Basic Results of the Geometry of Numbers
92(6)
5.3 Lattice Points Inside Circles
98(7)
5.4 Lattice Points on Circles
105(8)
5.5 Lattice Points Inside Spheres
113(5)
5.6 Lattice Points on Spheres
118(3)
6 Preparatory Tools of Mathematical Physics
121(102)
6.1 Integral Theorems for the Laplace Operator
122(11)
6.2 Integral Theorems for the Laplace-Beltrami Operator
133(6)
6.3 Tools Involving the Laplace Operator
139(5)
6.4 Radial and Angular Decomposition of Harmonics
144(36)
6.5 Integral Theorems for the Helmholtz-Beltrami Operator
180(12)
6.6 Radial and Angular Decomposition of Metaharmonics
192(23)
6.7 Tools Involving Helmholtz Operators
215(8)
7 Preparatory Tools of Fourier Analysis
223(24)
7.1 Periodical Polynomials and Fourier Expansions
224(3)
7.2 Classical Fourier Transform
227(2)
7.3 Poisson Summation and Periodization
229(3)
7.4 Gauß-Weierstraß and Abel-Poisson Transforms
232(11)
7.5 Hankel Transform and Discontinuous Integrals
243(4)
8 Lattice Function for the Iterated Helmholtz Operator
247(22)
8.1 Lattice Function for the Helmholtz Operator
248(7)
8.2 Lattice Function for the Iterated Helmholtz Operator
255(1)
8.3 Lattice Function in Terms of Circular Harmonics
256(9)
8.4 Lattice Function in Terms of Spherical Harmonics
265(4)
9 Euler Summation on Regular Regions
269(34)
9.1 Euler Summation Formula for the Iterated Laplace Operator
270(8)
9.2 Lattice Point Discrepancy Involving the Laplace Operator
278(4)
9.3 Zeta Function and Lattice Function
282(12)
9.4 Euler Summation Formulas for Iterated Helmholtz Operators
294(5)
9.5 Lattice Point Discrepancy Involving the Helmholtz Operator
299(4)
10 Lattice Point Summation
303(20)
10.1 Integral Asymptotics for (Iterated) Lattice Functions
304(4)
10.2 Convergence Criteria and Theorems
308(4)
10.3 Lattice Point-Generated Poisson Summation Formula
312(2)
10.4 Classical Two-Dimensional Hardy-Landau Identity
314(3)
10.5 Multi-Dimensional Hardy-Landau Identities
317(6)
11 Lattice Ball Summation
323(20)
11.1 Lattice Ball-Generated Euler Summation Formulas
324(4)
11.2 Lattice Ball Discrepancy Involving the Laplacian
328(3)
11.3 Convergence Criteria and Theorems
331(6)
11.4 Lattice Ball-Generated Poisson Summation Formula
337(1)
11.5 Multi-Dimensional Hardy-Landau Identities
338(5)
12 Poisson Summation on Regular Regions
343(18)
12.1 Theta Function and Gauß-Weierstraß Summability
344(6)
12.2 Convergence Criteria for the Poisson Series
350(5)
12.3 Generalized Parseval Identity
355(4)
12.4 Minkowski's Lattice Point Theorem
359(2)
13 Poisson Summation on Planar Regular Regions
361(24)
13.1 Fourier Inversion Formula
362(3)
13.2 Weighted Two-Dimensional Lattice Point Identities
365(14)
13.3 Weighted Two-Dimensional Lattice Ball Identities
379(6)
14 Planar Distribution of Lattice Points
385(44)
14.1 Qualitative Hardy-Landau Induced Geometric Interpretation
386(5)
14.2 Constant Weight Discrepancy
391(5)
14.3 Almost Periodicity of the Constant Weight Discrepancy
396(10)
14.4 Angular Weight Discrepancy
406(2)
14.5 Almost Periodicity of the Angular Weight Discrepancy
408(1)
14.6 Radial and Angular Weights
409(6)
14.7 Non-Uniform Distribution of Lattice Points
415(6)
14.8 Quantitative Step Function Oriented Geometric Interpretation
421(8)
15 Conclusions
429(2)
15.1 Summary
429(1)
15.2 Outlook
430(1)
Bibliography 431(12)
Index 443
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.
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