Abstract
A variety of constructions of rational approximations of orthogonal expansions has been discussed in the series of works of 1960–1970 (see [H], [F], [CL], [Gr], and also the monograph of G.A. Baker, Jr. and P. Graves-Morris [BG, Part 2, §1.6]). The greatest interest relates to the definitions of rational approximants which extend the basic definitions (in the sense of Padé-Baker and Frobenius) of the classical Padé approximants of power series to the case of series in orthogonal polynomials. In contrast to the classical case, these definitions lead to substantially different rational approximants of orthogonal expansions. The problems of convergence of the rows of the corresponding Padé tables have been investigated by S. Suetin [S2], [S3], and [Si]. The main results of the present article concern the diagonal Padé approximants of orthogonal expansions. Our purpose is to investigate the rate of convergence of these approximants for Markov type functions.
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References
Akhiezer, N.I., Lectures on Theory of Approximations, 2nd ed., “Nauka”, Moscow, 1965; English transl. of 1st ed.: Ungar, New York, 1956.
Baker, George A. Jr., and Graves-Morris, Peter, Padé Approximants, Part I: Basic Theory. Part II: Extensions and Applications. Encyclopedia of Mathematics and its Applications, 13 & 14, Cambridge University Press, Cambridge, 1981.
Clenshaw, C.W., and Lord, K., Rational approximations from Chebyshev series, In: Studies in Numerical Analysis, Academic Press, London, (1974), 95–113.
Fleischer, J., Nonlinear Padé approximants for Legendre series, J. of Math, and Physics, 14, No. 2, (1973), 246–248.
Gonchar, A.A., On the degree of rational approximation of analytic functions, Trudy Math. Inst. Steklov, 166 (1984), 52–60; English transl. in Proc. Steklov Math. Inst., 166, No. 1, (1986), 53–61.
Gonchar, A.A., On the speed of rational approximation of some analytic functions, Math. Sb., 105 (147) (1978), 147–163; English transl. in Math. USSR Sb., 34, No. 2, (1978), 131–146.
Gonchar, A.A., and Lopez-Lagomasino, Guillermo, On Markov’s theorem for multipoint Padé approximants, Mat. Sb. 105 (147) (1978), 511–524; English transl. in Math. USSR Sb., 34, No. 4, (1978), 449–460.
Gonchar, A.A., and Rakhmanov, E.A., Equilibrium distributions and degree of rational approximation of analytic functions, Mat. Sb., 134 (176) (1987), 306–352; English transl. in Math. USSR Sb., 62, No. 2, (1989), 305–348.
Gonchar, A.A., and Rakhmanov, E.A., The equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb., 125 (167) (1984), 117–127; English transl. in Math. USSR Sb., 53, No. 1, (1986), 119–130.
Gragg, W.B., Laurent, Fourier and Chebyshev Padé tables, In: Padé and Rational Approximation, (Eds. E.B. Saff and R.S. Varga), Academic Press, New York (1977), 61–70.
Holdeman, J.T. Jr., A method for approximation of functions defined by formal series expansion in orthogonal polynomials, Math. Comp., 23, No. 106, (1969), 275–287.
Landkof, N.S., Foundations of Modern Potential Theory, “Nauka”, Moscow, 1966; English transl., Springer-Verlag, Berlin, 1972.
Suetin, S.P., Inverse theorem on generalized Padé approximants, Math. Sb., 109 (151) (1979), 629–646; English transl. in Math. USSR Sb, 37, No. 4, (1980), 581–597.
Suetin, S.P., On the convergence of rational approximants to the polynomial expansions in the domains of meromorphy of a given function, Math. Sb., 105 (147) (1978), 413–430; English transl. in Math. USSR Sb, 34, No. 3, (1978), 367–381.
Suetin, S.P., On Montessus de Ballore’s theorem for rational approximants of orthogonal expansions, Math. Sb, 114 (156) (1981), 451–464; English transl. in Math. USSR. Sb., 42 (1982).
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Gonchar, A.A., Rakhmanov, E.A., Suetin, S.P. (1992). On the Rate of Convergence of Padé Approximants of Orthogonal Expansions. In: Gonchar, A.A., Saff, E.B. (eds) Progress in Approximation Theory. Springer Series in Computational Mathematics, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2966-7_7
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