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Matrix Riemann–Hilbert problems for Pade approximants of orthogonal expansions

A. I. Aptekarev, A. I. Bogolyubsky


Abstract: The Markov-type functions generated by measures given on some interval are considered. We are constructing the Pade approximants of orthogonal expansions for their Fourier series expansion by orthogonal polynomials on some other interval. Besides, we are studying both types of such constructions: linear Frobenius–Pade approximants and nonlinear Fourier–Pade ones. We have obtained two main new results in this paper: complete set of orthogonality relations for Fourier–Pade approximants denominators, and also equivalent reformulation of the problems concerning Pade–Fourier approximants of orthogonal expansions in terms of matrix Riemann–Hilbert problems.

Keywords: Pade–Chebyshev approximants, Pade approximants of orthogonal expansions, orthogonal polynomials, Markov-type functions, matrix Riemann–Hilbert problem.