We study the Pade approximations of first and second kinds of families of so-called Lerch functions. They can be efficiently determined by using ideas of Riemann and Chudnovsky on the monodromy of rigid systems of dif...
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We study the Pade approximations of first and second kinds of families of so-called Lerch functions. They can be efficiently determined by using ideas of Riemann and Chudnovsky on the monodromy of rigid systems of differential equations. In addition, we give some applications to diophantine approximations. (C) 2012 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG).
The conference "approximation and extrapolation of convergent and divergent sequences and series" discusses some very old problems of mathematical analysis and modern approaches for their solution based on s...
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The conference "approximation and extrapolation of convergent and divergent sequences and series" discusses some very old problems of mathematical analysis and modern approaches for their solution based on so-called nonlinear sequence transformations. In spite of all the advances in computer hard and software, slowly convergent or divergent sequences and series are still annoying obstacles not only in mathematics, but in particular also in other mathematically oriented disciplines. This articles tries to give a highly condensed survey of the topics treated at this conference. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.
The conference "approximation and extrapolation of convergent and divergent sequences and series" discusses some very old problems of mathematical analysis and modern approaches for their solution based on s...
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The conference "approximation and extrapolation of convergent and divergent sequences and series" discusses some very old problems of mathematical analysis and modern approaches for their solution based on so-called nonlinear sequence transformations. In spite of all the advances in computer hard and software, slowly convergent or divergent sequences and series are still annoying obstacles not only in mathematics, but in particular also in other mathematically oriented disciplines. This articles tries to give a highly condensed survey of the topics treated at this conference. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.
The authors deal with a class of rationalapproximation operators and their degree of approximation and analytic-preserving property, derive the expression of the form for the error in approximation of the operato...
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ISBN:
(纸本)9781424470815;9780769540474
The authors deal with a class of rationalapproximation operators and their degree of approximation and analytic-preserving property, derive the expression of the form for the error in approximation of the operator, and give the results on the analyticity of the operators.
We consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coef...
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We consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coefficients. Our results are based on several observations and algorithms in [G. Beylkin, L. Monzon, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (1) (2005) 17-48]. We avoid both the Gibbs phenomenon and the use of windows or filtering by constructing approximations to the available Fourier data via a short sum of decaying exponentials. Using these exponentials, we extrapolate the Fourier data to the whole real line and, on taking the inverse Fourier transform, obtain an efficient rational representation in the spatial domain. An important feature of this rational representation is that the positions of its poles indicate location of singularities of the function. We consider these representations in the absence of noise and discuss the impact of adding white noise to the Fourier data. We also compare our results with those obtained by other techniques. As an example of application, we consider our approach in the context of the kernel polynomial method for estimating density of states (eigenvalues) of Hermitian operators. We briefly consider the related problem of approximation by rational functions and provide numerical examples using our approach. (C) 2009 Elsevier Inc. All rights reserved.
We present some rationalapproximations of the sign function and analyze their convergence. The rate of convergence is shown to increase with the degree of the denominator of the rationalapproximation. Several numeri...
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We present some rationalapproximations of the sign function and analyze their convergence. The rate of convergence is shown to increase with the degree of the denominator of the rationalapproximation. Several numerical tests are presented.
Our goal is to survey, using three examples, how high-precision computations have stimulated mathematical research in the areas of polynomial and rationalapproximation theory. The first example will be the ''...
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Our goal is to survey, using three examples, how high-precision computations have stimulated mathematical research in the areas of polynomial and rationalapproximation theory. The first example will be the ''1/9'' Conjecture in rationalapproximation theory. Here high-precision computations gave strong evidence that this conjecture is false. Gonchar and Rakhmanov have given an exact solution of this conjecture. The second example will be the ''8'' Conjecture in rationalapproximation theory. In this case, high-precision computations and the use of the Richardson extrapolation method led to this conjecture. Stahl has proved that this conjecture and its generalization are true. The final example will be the Bernstein Conjecture in polynomial approximation theory and its generalization.
In computing best min-max rationalapproximations by the second algorithm of Remez (which is an iterative procedure), one must provide a starting approximation. A method proposed by Ralston and one by Werner are shown...
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For any compact set X, let C(X) denote the continuous functions on X and R(X) the functions on X which are uniformly approximate by rationalfunctions with poles off X. Let A denote a subnormal operator having no redu...
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