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This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.

Diophantine equations. --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Diophantic equations --- Equations, Diophantic --- Equations, Diophantine --- Equations, Indefinite --- Equations, Indeterminate --- Indefinite equations --- Indeterminate equations --- Diophantine analysis --- Algebraic Number Theory. --- Computational Method. --- Diophantine Geometry. --- Elliptic Diophantine Equation. --- Magma.

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Lectures on Constructive Approximation: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball focuses on spherical problems as they occur in the geosciences and medical imaging. It comprises the author’s lectures on classical approximation methods based on orthogonal polynomials and selected modern tools such as splines and wavelets. Methods for approximating functions on the real line are treated first, as they provide the foundations for the methods on the sphere and the ball and are useful for the analysis of time-dependent (spherical) problems. The author then examines the transfer of these spherical methods to problems on the ball, such as the modeling of the Earth’s or the brain’s interior. Specific topics covered include: * the advantages and disadvantages of Fourier, spline, and wavelet methods * theory and numerics of orthogonal polynomials on intervals, spheres, and balls * cubic splines and splines based on reproducing kernels * multiresolution analysis using wavelets and scaling functions This textbook is written for students in mathematics, physics, engineering, and the geosciences who have a basic background in analysis and linear algebra. The work may also be suitable as a self-study resource for researchers in the above-mentioned fields.

Approximation theory. --- Numerical analysis. --- Mathematical analysis --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Mathematics. --- Functions, special. --- Fourier analysis. --- Mathematical physics. --- Approximations and Expansions. --- Special Functions. --- Fourier Analysis. --- Mathematical Methods in Physics. --- Numerical Analysis. --- Physical mathematics --- Physics --- Analysis, Fourier --- Special functions --- Math --- Science --- Mathematics --- Approximation theory --- Fourier analysis --- Numerical analysis --- Spherical functions --- Spline theory --- Wavelets (Mathematics) --- 519.65 --- 681.3*G12 --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 519.65 Approximation. Interpolation --- Approximation. Interpolation --- Wavelet analysis --- Harmonic analysis --- Spline functions --- Interpolation --- Functions, Spherical --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Special functions. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics