Abstract:
Let ˆσˆσ be a Cauchy transform of a possibly complex-valued Borel measure σσ and {pn}{pn} a system of orthonormal polynomials with respect to a measure μμ, where supp(μ)∩supp(σ)=∅. An (m,n)th Frobenius-Padé approximant to ˆσ is a rational function P/Q, deg(P)≤m, deg(Q)≤n, such that the first m+n+1 Fourier coefficients of the remainder function Qˆσ−P vanish when the form is developed into a series with respect to the polynomials pn. We investigate the convergence of the Frobenius-Padé approximants to ˆσ along ray sequences n/(n+m+1)→c>0, n−1≤m, when μ and σ are supported on intervals of the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the corresponding interval are holomorphic functions.
Bibliography: 30 titles.
Keywords:
Frobenius-Padé approximants, linear Padé-Chebyshev approximants, Padé approximants of orthogonal expansions, orthogonality, Markov-type functions, Riemann-Hilbert matrix problem.
A. I. Aptekarev's research was supported by the Russian Science Foundation (grant no. 14-21-00025). A. I. Bogolyubskii's research was supported by the Russian Foundation for Basic Research (grant nos. 14-01-00604_а and 17-01-00614_a). M. L. Yattselev's research was supported by the Simons Foundation (grant #354538).
Citation:
A. I. Aptekarev, A. I. Bogolyubskii, M. Yattselev, “Convergence of ray sequences of Frobenius-Padé approximants”, Sb. Math., 208:3 (2017), 313–334
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This publication is cited in the following 13 articles:
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N. R. Ikonomov, S. P. Suetin, “On some potential-theoretic problems related to the asymptotics of Hermite–Padé polynomials”, Sb. Math., 215:8 (2024), 1053–1064
A. I. Aptekarev, S. A. Denisov, M. L. Yattselev, “Jacobi matrices on trees generated by Angelesco systems: asymptotics of coefficients and essential spectrum”, J. Spectr. Theory, 11:4 (2021), 1511–1597
N. R. Ikonomov, S. P. Suetin, “Scalar Equilibrium Problem and the Limit Distribution of Zeros of Hermite–Padé Polynomials of Type II”, Proc. Steklov Inst. Math., 309 (2020), 159–182
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S. P. Suetin, “Existence of a three-sheeted Nutall surface for a certain class of infinite-valued analytic functions”, Russian Math. Surveys, 74:2 (2019), 363–365
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S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials for a complex Nikishin system”, Russian Math. Surveys, 73:2 (2018), 363–365
G. López Lagomasino, W. Van Assche, “Riemann-Hilbert analysis for a Nikishin system”, Sb. Math., 209:7 (2018), 1019–1050
S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padé polynomials for a Nikishin system”, Proc. Steklov Inst. Math., 301 (2018), 245–261
A. P. Starovoitov, “Hermite–Padé approximants of the Mittag-Leffler functions”, Proc. Steklov Inst. Math., 301 (2018), 228–244
S. P. Suetin, “On an Example of the Nikishin System”, Math. Notes, 104:6 (2018), 905–914
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