Multipoint Hermite–Padé approximants for three beta functions

Sorokin, V.

doi:10.1090/mosc/276

Multipoint Hermite–Padé approximants for three beta functions
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by V. N. Sorokin
Translated by: A. Alimov

Trans. Moscow Math. Soc. 2018, 117-134

DOI: https://doi.org/10.1090/mosc/276

Published electronically: November 29, 2018

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Abstract:

This paper is concerned with joint multipoint rational approximants with a common denominator for three beta functions. The limit distributions of the zeros of the denominators are obtained in terms of equilibrium logarithmic potentials and in terms of meromorphic functions on Riemann surfaces.

References

A. A. Kandayan, Multipoint Padé approximants of the beta function, Mat. Zametki 85 (2009), no. 2, 189–203 (Russian, with Russian summary); English transl., Math. Notes 85 (2009), no. 1-2, 176–189. MR 2547999, DOI 10.1134/S0001434609010210
A. A. Kandayan and V. N. Sorokin, Multipoint Hermite-Padé approximants of beta functions, Mat. Zametki 87 (2010), no. 2, 217–232 (Russian, with Russian summary); English transl., Math. Notes 87 (2010), no. 1-2, 204–217. MR 2731473, DOI 10.1134/S000143461001027X
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, Translations of Mathematical Monographs, vol. 92, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas. MR 1130396, DOI 10.1090/mmono/092
V. N. Sorokin, Polynomials of joint orthogonality for discrete Meixner measures, Mat. Sb. 201 (2010), no. 10, 137–160 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 9-10, 1539–1561. MR 2768827, DOI 10.1070/SM2010v201n10ABEH004121
A. A. Gonchar and E. A. Rakhmanov, On the convergence of simultaneous Padé approximants for systems of functions of Markov type, Trudy Mat. Inst. Steklov. 157 (1981), 31–48, 234 (Russian). Number theory, mathematical analysis and their applications. MR 651757
A. A. Gonchar and E. A. Rakhmanov, The equilibrium measure and distribution of zeros of extremal polynomials, Mat. Sb. (N.S.) 125(167) (1984), no. 1(9), 117–127 (Russian). MR 760416
A. A. Gonchar and E. A. Rakhmanov, The equilibrium problem for vector potentials, Uspekhi Mat. Nauk 40 (1985), no. 4(244), 155–156 (Russian). MR 807734
E. A. Rakhmanov, Uniform measure and distribution of zeros of extremal polynomials of a discrete variable, Mat. Sb. 187 (1996), no. 8, 109–124 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 8, 1213–1228. MR 1418343, DOI 10.1070/SM1996v187n08ABEH000153
A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, Padé-Chebyshev approximants for multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets, Uspekhi Mat. Nauk 66 (2011), no. 6(402), 3–36 (Russian, with Russian summary); English transl., Russian Math. Surveys 66 (2011), no. 6, 1015–1048. MR 2963450, DOI 10.1070/RM2011v066n06ABEH004769
A. Martines-Finkel′shteĭn, E. A. Rakhmanov, and S. P. Suetin, Variation of equilibrium energy and the $S$-property of a stationary compact set, Mat. Sb. 202 (2011), no. 12, 113–136 (Russian, with Russian summary); English transl., Sb. Math. 202 (2011), no. 11-12, 1831–1852. MR 2919251, DOI 10.1070/SM2011v202n12ABEH004209
E. M. Nikišin, Simultaneous Padé approximants, Mat. Sb. (N.S.) 113(155) (1980), no. 4(12), 499–519, 637 (Russian). MR 602272
E. M. Nikishin, Asymptotic behavior of linear forms for simultaneous Padé approximants, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (1986), 33–41, 84 (Russian). MR 842319
A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin, On Hermite-Padé approximants for systems of functions of Markov type, Mat. Sb. 188 (1997), no. 5, 33–58 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 5, 671–696. MR 1478629, DOI 10.1070/SM1997v188n05ABEH000225