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This monograph is concerned with wavelet harmonic analysis on the sphere. By starting with orthogonal polynomials and functional Hilbert spaces on the sphere, the foundations are laid for the study of spherical harmonics such as zonal functions. The book also discusses the construction of wavelet bases using special functions, especially Bessel, Hermite, Tchebychev, and Gegenbauer polynomials. ContentsReview of orthogonal polynomialsHomogenous polynomials and spherical harmonicsReview of special functionsSpheroidal-type wavelets Some applicationsSome applications

Wavelets (Mathematics) --- Wavelet analysis --- Harmonic analysis --- Wavelets. --- harmonic analysis. --- special functions. --- spherical harmonics. --- zonal functions.

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This book provides a comprehensive presentation of geometric results, primarily from the theory of convex sets, that have been proved by the use of Fourier series or spherical harmonics. An important feature of the book is that all necessary tools from the classical theory of spherical harmonics are presented with full proofs. These tools are used to prove geometric inequalities, stability results, uniqueness results for projections and intersections by hyperplanes or half-spaces and characterisations of rotors in convex polytopes. Again, full proofs are given. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets. This treatise will be welcomed both as an introduction to the subject and as a reference book for pure and applied mathematics.

Convex sets. --- Fourier series. --- Spherical harmonics. --- Functions, Potential --- Potential functions --- Harmonic analysis --- Harmonic functions --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Calculus --- Fourier analysis --- Sets, Convex --- Convex domains --- Set theory

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"The purpose of this monograph is to discuss recent developments in the analysis of isotropic spherical random fields, with a view towards applications in Cosmology.We shall be concerned in particular with the interplay among three leading themes, namely: - the connection between isotropy, representation of compact groups and spectral analysis for random fields, including the characterization of polyspectra and their statistical estimation - the interplay between Gaussianity, Gaussian subordination, nonlinear statistics, and recent developments in the methods of moments and diagram formulae to establish weak convergence results - the various facets of high-resolution asymptotics, including the high-frequency behaviour of Gaussian subordinated random fields and asymptotic statistics in the high-frequency sense"--

Spherical harmonics. --- Random fields. --- Compact groups. --- Cosmology --- Astronomy --- Deism --- Metaphysics --- Groups, Compact --- Locally compact groups --- Topological groups --- Fields, Random --- Stochastic processes --- Functions, Potential --- Potential functions --- Harmonic analysis --- Harmonic functions --- Statistical methods.

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These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.

Engineering & Applied Sciences --- Civil & Environmental Engineering --- Operations Research --- Applied Mathematics --- Spherical harmonics. --- Spherical functions. --- Functions, Spherical --- Functions, Potential --- Potential functions --- Mathematics. --- Approximation theory. --- Integral equations. --- Partial differential equations. --- Special functions. --- Numerical analysis. --- Physics. --- Numerical Analysis. --- Special Functions. --- Approximations and Expansions. --- Integral Equations. --- Partial Differential Equations. --- Physics, general. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Mathematical analysis --- Special functions --- Partial differential equations --- Equations, Integral --- Functional equations --- Functional analysis --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Harmonic analysis --- Harmonic functions --- Functions, special. --- Differential equations, partial.

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Collects the material developed by the Geomathematics Group, TU Kaiserslautern, to set up a theory of spherical functions of mathematical (geo-)physics. This work provides the palette of spherical (trial) functions for modeling and simulating phenomena and processes of the Earth system.

Fractals. --- Geology -- Mathematics. --- Spherical functions. --- Cosmic Physics --- Geology - General --- Physics --- Geology --- Physical Sciences & Mathematics --- Earth & Environmental Sciences --- Mathematics. --- Functions, Spherical --- Geomathematics --- Mathematical geology --- Earth sciences. --- Geophysics. --- Applied mathematics. --- Engineering mathematics. --- Earth Sciences. --- Geophysics/Geodesy. --- Applications of Mathematics. --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Physical geography. --- Math --- Science --- Geography --- Engineering --- Engineering analysis --- Mathematical analysis --- Geological physics --- Terrestrial physics --- Earth sciences --- Mathematics