On the Rate of Convergence of Padé Approximants of Orthogonal Expansions
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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