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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

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While most textbooks on Numerical Analysis discuss linear techniques for the solution of various numerical problems, this book introduces and illustrates nonlinear methods. It presents several nonlinear techniques resulting mainly from the use of Padé approximants and rational interpolants.

Analyse numérique --- Numerical analysis --- Numerieke analyse --- 517.518.8 --- 519.6 --- 681.3*G11 --- 681.3*G12 --- Mathematical analysis --- Approximation of functions by polynomials and their generalizations --- Computational mathematics. Numerical analysis. Computer programming --- Interpolation: difference formulas; extrapolation; smoothing; spline and piecewise polynomial interpolation (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) --- 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) --- 681.3*G11 Interpolation: difference formulas; extrapolation; smoothing; spline and piecewise polynomial interpolation (Numerical analysis) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Analyse numérique --- ELSEVIER-B EPUB-LIV-FT --- Mathematical analysis. --- Numerical analysis. --- 517.1 Mathematical analysis --- Analyse numérique. --- Series (mathematique) --- Equations non lineaires --- Sommation --- Approximation des solutions

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Approximation theory and functional analysis

Functionaalanalyse. --- Functional analysis --- Approximation theory --- Analyse fonctionnelle --- Théorie de l'approximation --- Congresses. --- Congrès --- 51 --- -Functional analysis --- -517.988 --- 519.6 --- 681.3*G12 --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Mathematics --- Congresses --- Nonlinear functional analysis and approximation methods --- Computational mathematics. Numerical analysis. Computer programming --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 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) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 517.988 Nonlinear functional analysis and approximation methods --- 51 Mathematics --- 517.988 --- 51 Wiskunde. Mathematiek --- Wiskunde. Mathematiek --- Functional analysis - Congresses --- Approximation theory - Congresses

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Diagnosis and Management of Polycystic Ovary Syndrome (PCOS) is a clinical reference work for primary care physicians, internists, general endocrinologists, obstetricians, gynecologists and students. PCOS is a common but often misdiagnosed disease. Many symptoms can be alleviated by early intervention and effective management.Prominent endocrinologists have contributed recent data current research on the pathogenesis, manifestations, diagnosis and treatment of PCOS. The variety of medical issues presenting in PCOS patients result in late referrals or inappropriate advice. This title will be a tool in understanding the metabolic and genetic basis of PCOS, while providing management strategies.

Pade ́ approximant --Congresses. --- Polycystic ovary syndrome --- Hyperandrogenism --- Ovarian Cysts --- Analytical, Diagnostic and Therapeutic Techniques and Equipment --- Ovarian Diseases --- Cysts --- Gonadal Disorders --- Neoplasms --- Adnexal Diseases --- Endocrine System Diseases --- Diseases --- Genital Diseases, Female --- Female Urogenital Diseases --- Female Urogenital Diseases and Pregnancy Complications --- Polycystic Ovary Syndrome --- Diagnosis --- Medicine --- Health & Biological Sciences --- Gynecology & Obstetrics --- Clinical Endocrinology --- Polycystic ovary syndrome. --- Gynecology. --- Gynaecology --- PCOD (Gynecology) --- PCOS (Gynecology) --- Polycystic ovarian disease --- Polyfollicular ovarian disease --- Sclerocystic ovarian degeneration --- Sclerocystic ovaries --- Sclerocystic ovary syndrome --- Stein-Leventhal syndrome --- Approximation: chebyshev elementary function least squares linear approximation minimax approximation and algorithms nonlinear and rational approximation spline and piecewise polynomial approximation (Numerical analysis) --- Padé approximant --- 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) --- 530.12 <063> --- Relativity principle--Congressen --- -Relativity principle--Congressen --- 530.12 <063> Relativity principle--Congressen --- Anniversaries, etc. --- Medicine. --- General practice (Medicine). --- Endocrinology. --- Medicine & Public Health. --- General Practice / Family Medicine. --- 517.518.8 --- 517.52 --- 519.6 --- 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) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- 517.52 Series and sequences --- Series and sequences --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Approximation of functions by polynomials and their generalizations --- Physics --- Congresses --- Einstein, Albert, --- Numerical approximation theory --- Generative organs, Female --- Einstein, Albert --- Ovaries --- Syndromes --- Congresses. --- Physique --- Congrès --- Family medicine. --- Internal medicine --- Hormones --- Family practice (Medicine) --- General practice (Medicine) --- Physicians (General practice) --- Analyse numérique. --- Numerical analysis --- Aiyinsitan, Abote, --- Aĭnshtaĭn, Albert, --- Ainshutain, A, --- Ain̲sṭain̲, Ālparṭ, --- Ainsṭāina, Albarṭa, --- Ajnštajn, Albert, --- Āynishtayn, --- Aynshtayn, Albert, --- Eĭnshteĭn, Alʹbert, --- אינשטין, אלברט, --- איינשטיין --- איינשטיין, אלבערט, --- איינשטיין, אלברט --- איינשטיין, אלברט, --- Aynştayn, Elbêrt, --- Īnshtīn, --- Aynîştayn, --- Aiyinsitan, --- 愛因斯坦, --- 爱因斯坦, --- General relativity (Physics) --- Analyse numérique --- Numerical analysis. --- Approximation et developpements --- Approximation de pade

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Evolving from an elementary discussion, this book develops the Euclidean algorithm to a very powerful tool to deal with general continued fractions, non-normal Padé tables, look-ahead algorithms for Hankel and Toeplitz matrices, and for Krylov subspace methods. It introduces the basics of fast algorithms for structured problems and shows how they deal with singular situations. Links are made with more applied subjects such as linear system theory and signal processing, and with more advanced topics and recent results such as general bi-orthogonal polynomials, minimal Padé approximation, poly

Ordered algebraic structures --- Numerical approximation theory --- Computer science --- lineaire algebra --- Algebras, Linear --- Euclidean algorithm --- Orthogonal polynomials --- Padé approximant --- #TELE:SISTA --- 519.6 --- 681.3*G11 --- 681.3*G12 --- 681.3*G13 --- Algorithm of Euclid --- Continued division --- Division, Continued --- Euclid algorithm --- Euclidian algorithm --- Euclid's algorithm --- Algorithms --- Number theory --- Linear algebra --- Algebra, Universal --- Generalized spaces --- Mathematical analysis --- Calculus of operations --- Line geometry --- Topology --- 681.3*G13 Numerical linear algebra: conditioning; determinants; Eigenvalues; error analysis; linear systems; matrix inversion; pseudoinverses; sparse and very largesystems --- Numerical linear algebra: conditioning; determinants; Eigenvalues; error analysis; linear systems; matrix inversion; pseudoinverses; sparse and very largesystems --- 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) --- 681.3*G11 Interpolation: difference formulas; extrapolation; smoothing; spline and piecewise polynomial interpolation (Numerical analysis) --- Interpolation: difference formulas; extrapolation; smoothing; spline and piecewise polynomial interpolation (Numerical analysis) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- Fourier analysis --- Functions, Orthogonal --- Polynomials --- Approximant, Padé --- Approximation theory --- Continued fractions --- Power series --- Euclidean algorithm. --- Algebras, Linear. --- Padé approximant. --- Orthogonal polynomials. --- Padé approximant. --- Pade approximant.