Bosuwan, Nattapong A note on the rate of convergence of poles of generalized Hermite-Padé approximants. (English) Zbl 1463.30175 Thai J. Math., Spec. Iss.: Annual Meeting in Mathematics 2019, 25-37 (2020). Summary: We consider row sequences of three generalized Hermite-Padé approximations (orthogonal Hermite-Padé approximation, Hermite-Padé-Faber approximation, and multipoint Hermite-Padé approximation) of a vector of the approximated functions \(\mathbf{F}\) and prove that if \(\mathbf{F}\) has a system pole of order \(\nu \), then such system pole attracts at least \(\nu\) zeros of denominators of these approximants at the rate of a geometric progression. Moreover, the rates of these attractions are estimated. MSC: 30E10 Approximation in the complex plane 41A21 Padé approximation 41A25 Rate of convergence, degree of approximation Keywords:orthogonal polynomials; Faber polynomials; interpolation; Hermite-Padé approximation; rate of convergence PDFBibTeX XMLCite \textit{N. Bosuwan}, Thai J. Math., 25--37 (2020; Zbl 1463.30175) Full Text: Link References: [1] A.A. Gonchar, Poles of rows of the Pad´e table and meromorphic continuation of functions, Sb. Math. 43 (1981) 527-8-546. · Zbl 0492.30020 [2] N. Bosuwan, G. L´opez Lagomasino, Determining system poles using row sequences of orthogonal Hermite-Pad´e approximants, J. Approx. Theory 231 (2018) 15-40. · Zbl 1392.41010 [3] N. Bosuwan, G. L´opez Lagomasino, Direct and inverse results on row sequences of simultaneous Pad´e-Faber approximants, Mediterr J. Math. 16 (36) (2019) https://doi:10.1007/s00009-019-1307-0. · Zbl 1417.30029 [4] N. Bosuwan, G. L´opez Lagomasino, Y. Zaldivar Gerpe, Direct and inverse results for multipoint Hermite-Pad´e approximants, Anal. Math. Phys. Accepted. · Zbl 1426.30024 [5] E. Mi˜na-D´ıaz, An expansion for polynomials orthogonal over an analytic Jordan curve, Comm. Math. Phys. 285 (2009) 1109-1128. · Zbl 1180.30042 [6] N. Stylianopoulos, Strong asymptotics for Bergman polynomials over domains with corners and applications, Constr. Approx. 38 (2013) 59-100. · Zbl 1283.30006 [7] H. Stahl, V. Totik, General Orthogonal Polynomials, Cambridge University Press, Vol. 43, Cambridge, 1992. · Zbl 0791.33009 [8] W. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. Math. 3 (1969) 127-232. · Zbl 0183.07503 [9] J. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.