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In this treatise, the authors present the general theory of orthogonal polynomials on the complex plane and several of its applications. The assumptions on the measure of orthogonality are general, the only restriction is that it has compact support on the complex plane. In the development of the theory the main emphasis is on asymptotic behaviour and the distribution of zeros. In the following chapters, the author explores the exact upper and lower bounds are given for the orthonormal polynomials and for the location of their zeros; regular n-th root asymptotic behaviour; and applications of the theory, including exact rates for convergence of rational interpolants, best rational approximants and non-diagonal Pade approximants to Markov functions (Cauchy transforms of measures). The results are based on potential theoretic methods, so both the methods and the results can be extended to extremal polynomials in norms other than L2 norms. A sketch of the theory of logarithmic potentials is given in an appendix.

Orthogonal polynomials. --- Fourier analysis. --- Analysis, Fourier --- Mathematical analysis --- Fourier analysis --- Functions, Orthogonal --- Polynomials

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Harmonic analysis. Fourier analysis --- Mathematical physics --- Fourier analysis --- Analyse de Fourier --- Fourier analysis. --- Fourier Analysis. --- Fourier transformations --- Fourier series --- Functions, Special --- Laplace transformation --- Fourier, Transformations de --- Fonctions spéciales --- Transformation de Laplace --- Fourier, Séries de --- Fourier, Analyse de --- Fourier, Transformations de. --- Fonctions spéciales. --- Transformation de Laplace. --- Fourier, Séries de. --- Fourier, Analyse de. --- Fourier Analysis

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Orthogonal polynomials. --- Orthogonal polynomials --- 517.518.8 --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Approximation of functions by polynomials and their generalizations --- Fourier analysis --- Functions, Orthogonal --- Polynomials

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The most comprehensive treatment of FFTs to date. Van Loan captures the interplay between mathematics and the design of effective numerical algorithms--a critical connection as more advanced machines become available. A stylized Matlab notation, which is familiar to those engaged in high-performance computing, is used. The Fast Fourier Transform (FFT) family of algorithms has revolutionized many areas of scientific computation. The FFT is one of the most widely used algorithms in science and engineering, with applications in almost every discipline. This volume is essential for professionals interested in linear algebra as well as those working with numerical methods. The FFT is also a great vehicle for teaching key aspects of scientific computing.

Numerical analysis --- Fourier transformations --- Transformations de Fourier --- #TELE:SISTA --- #TELE:MI2 --- 517.518.8 --- 519.6 --- 681.3*F21 --- 681.3*G4 --- Transformations, Fourier --- Transforms, Fourier --- Fourier analysis --- Transformations (Mathematics) --- Approximation of functions by polynomials and their generalizations --- Computational mathematics. Numerical analysis. Computer programming --- Numerical algorithms and problems: computation of transforms; computations infinite fields; computations on matrices; computations on polynomials; numer-theoretic computations--See also {681.3*G1}; {681.3*G4}; {681.3*I1} --- Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- Fourier transformations. --- 681.3*G4 Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- 681.3*F21 Numerical algorithms and problems: computation of transforms; computations infinite fields; computations on matrices; computations on polynomials; numer-theoretic computations--See also {681.3*G1}; {681.3*G4}; {681.3*I1} --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Analyse numérique. --- Analyse numérique --- Numerical analysis. --- Analyse de fourier --- Fast fourier transform = fft