Approximation by amplitude and frequency operators

JOURNAL OF APPROXIMATION THEORY

卷:207

Article

Chunaev, Petr¹  Danchenko, Vladimir² 

[1] Univ Autonoma Barcelona, Fac Ciencies, Dept Matemat, Edifici C, Bellaterra 08193, Barcelona, Spain

[2] Vladimir State Univ, Funct Anal & Its Applicat Dept, Belokonskoy Str 3-7,Bldg 3, Vladimir 600000, Russia

关键词: Amplitude and frequency sum;    Discrete moment problem;    Regularization;    Interpolation;    Extrapolation;    Bessel functions;   

DOI  :  10.1016/j.jat.2016.02.005

来源: Elsevier

【 摘 要 】

We study Pade interpolation at the node z = 0 of functions f (z) = Sigma(infinity)(m=0)f(m)z(m), analytic in a neighbourhood of this node, by amplitude and frequency operators (sums) of the form Sigma(n)(k=1) mu(k)h(lambda(k)z), mu(k), lambda(k) is an element of C. Here h(z) = Sigma(infinity)(m=0) h(m)Z(m), h(m) not equal 0, is a fixed (basis) function, analytic at the origin, and the interpolation is carried out by an appropriate choice of amplitudes mu(k) and frequencies lambda(k). The solvability of the 2n-multiple interpolation problem is determined by the solvability of the associated moment problem Sigma(n)(k=1) mu(k)lambda(m)(k) = f(m)/h(m), m = (0, 2n-1) over bar. In a number of cases, when the moment problem is consistent, it can be solved by the classical method due to Prony and Sylvester, moreover, one can easily construct the corresponding interpolating sum too. In the case of inconsistent moment problems, we propose a regularization method, which consists in adding a special binomial c(1)z(n-1) + c(2)z(2n-1) to an amplitude and frequency sum so that the moment problem, associated with the sum obtained, can be already solved by the method of Prony and Sylvester. This approach enables us to obtain interpolation formulas with n nodes lambda(k)z, being exact for the polynomials of degree <= 2n - 1, whilst traditional formulas with the same number of nodes are usually exact only for the polynomials of degree <= n - 1. The regularization method is applied to numerical differentiation and extrapolation. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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