Transactions of the Moscow Mathematical Society

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Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces
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by N. R. Ikonomov and S. P. Suetin;
Translated by: the authors
Trans. Moscow Math. Soc. 2022, 33-54
DOI: https://doi.org/10.1090/mosc/344
Published electronically: September 23, 2024
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Abstract:

The structure of a Nuttall partition into sheets of some class of four-sheeted Riemann surfaces is studied. The corresponding class of multivalued analytic functions is a special class of algebraic functions of fourth order generated by the function inverse to the Zhukovskii function. We show that in this class of four-sheeted Riemann surfaces, the boundary between the second and third sheets of the Nuttall partition of the Riemann surface is completely characterized in terms of an extremal problem posed on the two-sheeted Riemann surface of the function $w$ defined by the equation $w^2=z^2-1$. In particular, we show that in this class of functions the boundary between the second and third sheets intersects neither the boundary between the first and second sheets nor that between the third and fourth sheets.
References
  • A. I. Aptekarev, Asymptotics of Hermite-Padé approximants for a pair of functions with branch points, Dokl. Akad. Nauk 422 (2008), no. 4, 443–445 (Russian); English transl., Dokl. Math. 78 (2008), no. 2, 717–719. MR 2475084, DOI 10.1134/S1064562408050207
  • A. I. Aptekarev, V. I. Buslaev, A. Martines-Finkel′shteĭn, and S. P. Suetin, Padé approximants, continued fractions, and orthogonal polynomials, Uspekhi Mat. Nauk 66 (2011), no. 6(402), 37–122 (Russian, with Russian summary); English transl., Russian Math. Surveys 66 (2011), no. 6, 1049–1131. MR 2963451, DOI 10.1070/RM2011v066n06ABEH004770
  • Alexander I. Aptekarev and Maxim L. Yattselev, Padé approximants for functions with branch points—strong asymptotics of Nuttall-Stahl polynomials, Acta Math. 215 (2015), no. 2, 217–280. MR 3455234, DOI 10.1007/s11511-016-0133-5
  • A. I. Aptekarev and D. N. Tulyakov, Nuttall’s Abelian integral on the Riemann surface of the cube root of a polynomial of degree 3, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 6, 5–42 (Russian, with Russian summary); English transl., Izv. Math. 80 (2016), no. 6, 997–1034. MR 3588811, DOI 10.4213/im8420
  • N. U. Arakelyan, Efficient analytic continuation of power series and the localization of their singularities, Izv. Nats. Akad. Nauk Armenii Mat. 38 (2003), no. 4, 5–24 (Russian, with English and Russian summaries); English transl., J. Contemp. Math. Anal. 38 (2003), no. 4, 2–20 (2004). MR 2133957
  • E. M. Chirka, Potentials on a compact Riemann surface, Tr. Mat. Inst. Steklova 301 (2018), no. Kompleksnyĭ Analiz, Matematicheskaya Fizika i Prilozheniya, 287–319 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 301 (2018), no. 1, 272–303. MR 3841675, DOI 10.1134/S0371968518020218
  • E. M. Chirka, Equilibrium measures on a compact Riemann surface, Tr. Mat. Inst. Steklova 306 (2019), no. Matematicheskaya Fisika i Prilozheniya, 313–351 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 306 (2019), no. 1, 296–334. MR 4040783, DOI 10.4213/tm4007
  • E. M. Chirka, Meromorphic interpolation on a compact Riemann surface, Mat. Zametki 106 (2019), no. 1, 154–157 (Russian); English transl., Math. Notes 106 (2019), no. 1-2, 156–159. MR 3981336, DOI 10.4213/mzm12317
  • E. M. Chirka, Capacities on a compact Riemann surface, Analysis and mathematical physics, Collected papers. On the occasion of the 70th birthday of Professor Armen Glebovich Sergeev, Tr. Mat. Inst. Steklova 311, ed. S. Yu. Nemirovski, A. V. Komlov, Steklov Math. Inst., Moscow, 2020, 281pp. (Russian).
  • A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and the rate of rational approximation of analytic functions, Mat. Sb. (N.S.) 134(176) (1987), no. 3, 306–352, 447 (Russian); English transl., Math. USSR-Sb. 62 (1989), no. 2, 305–348. MR 922628, DOI 10.1070/SM1989v062n02ABEH003242
  • 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
  • N. R. Ikonomov and S. P. Suetin, Scalar equilibrium problem and the limit distribution of zeros of Hermite-Padé polynomials of type II, Tr. Mat. Inst. Steklova 309 (2020), no. Sovremennye Problemy Matematicheskoĭ Teoreticheskoĭ Fiziki, 174–197 (Russian, with Russian summary). English version published in Proc. Steklov Inst. Math. 309 (2020), no. 1, 159–182. MR 4133451, DOI 10.4213/tm4080
  • A. V. Komlov, R. V. Pal′velev, S. P. Suetin, and E. M. Chirka, Hermite-Padé approximants for meromorphic functions on a compact Riemann surface, Uspekhi Mat. Nauk 72 (2017), no. 4(436), 95–130 (Russian, with Russian summary); English transl., Russian Math. Surveys 72 (2017), no. 4, 671–706. MR 3687129, DOI 10.4213/rm9786
  • Aleksandr Komlov, Polynomial Hermite Padé $m$-system and reconstruction of the values of algebraic functions, Extended abstracts fall 2019—spaces of analytic functions: approximation, interpolation, sampling, Trends Math. Res. Perspect. CRM Barc., vol. 12, Birkhäuser/Springer, Cham, [2021] ©2021, pp. 113–121. MR 4425816
  • Abey López-García and Guillermo López Lagomasino, Nikishin systems on star-like sets: ratio asymptotics of the associated multiple orthogonal polynomials, J. Approx. Theory 225 (2018), 1–40. MR 3733249, DOI 10.1016/j.jat.2017.10.002
  • E. Lopatin, On a generalization of the scalar approach to the problem of limit distribution of zeros of Hermite–Padé polynomials for a pair of functions forming the Nikishin system, Seminar on Complex Analysis (Gonchar Seminar), December 21, 2020.
  • Andrei Martínez-Finkelshtein, Evguenii A. Rakhmanov, and Sergey P. Suetin, Asymptotics of type I Hermite-Padé polynomials for semiclassical functions, Modern trends in constructive function theory, Contemp. Math., vol. 661, Amer. Math. Soc., Providence, RI, 2016, pp. 199–228. MR 3489559, DOI 10.1090/conm/661/13283
  • J. Nuttall and S. R. Singh, Orthogonal polynomials and Padé approximants associated with a system of arcs, J. Approximation Theory 21 (1977), no. 1, 1–42. MR 487173, DOI 10.1016/0021-9045(77)90117-4
  • J. Nuttall, Asymptotics of diagonal Hermite-Padé polynomials, J. Approx. Theory 42 (1984), no. 4, 299–386. MR 769985, DOI 10.1016/0021-9045(84)90036-4
  • E. A. Rakhmanov, Orthogonal polynomials and $S$-curves, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., vol. 578, Amer. Math. Soc., Providence, RI, 2012, pp. 195–239. MR 2964146, DOI 10.1090/conm/578/11484
  • E. A. Rakhmanov and S. P. Suetin, Distribution of zeros of Hermite-Padé polynomials for a pair of functions forming a Nikishin system, Mat. Sb. 204 (2013), no. 9, 115–160 (Russian, with Russian summary); English transl., Sb. Math. 204 (2013), no. 9-10, 1347–1390. MR 3137137, DOI 10.1070/sm2013v204n09abeh004343
  • E. A. Rakhmanov, Zero distribution for Angelesco Hermite-Padé polynomials, Uspekhi Mat. Nauk 73 (2018), no. 3(441), 89–156 (Russian, with Russian summary); English transl., Russian Math. Surveys 73 (2018), no. 3, 457–518. MR 3807896, DOI 10.4213/rm9832
  • Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, NJ, 1954. MR 65652
  • V. N. Sorokin, Hermite-Padé approximants to the Weyl function and its derivative for discrete measures, Mat. Sb. 211 (2020), no. 10, 139–156 (Russian, with Russian summary); English transl., Sb. Math. 211 (2020), no. 10, 1486–1503. MR 4153721, DOI 10.4213/sm8634
  • Herbert Stahl, Asymptotics of Hermite-Padé polynomials and related convergence results—a summary of results, Nonlinear numerical methods and rational approximation (Wilrijk, 1987) Math. Appl., vol. 43, Reidel, Dordrecht, 1988, pp. 23–53. MR 1005350
  • Herbert Stahl, The convergence of Padé approximants to functions with branch points, J. Approx. Theory 91 (1997), no. 2, 139–204. MR 1484040, DOI 10.1006/jath.1997.3141
  • Herbert R. Stahl, Sets of minimal capacity and extremal domains, arXiv: 1205.3811.
  • S. P. Suetin, On the uniform convergence of diagonal Padé approximants for hyperelliptic functions, Mat. Sb. 191 (2000), no. 9, 81–114 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 9-10, 1339–1373. MR 1805599, DOI 10.1070/SM2000v191n09ABEH000508
  • S. P. Suetin, On an example of the Nikishin system, Mat. Zametki 104 (2018), no. 6, 918–929 (Russian, with Russian summary); English transl., Math. Notes 104 (2018), no. 5-6, 905–914. MR 3881781, DOI 10.4213/mzm12181
  • S. P. Suetin, On a new approach to the problem of distribution of zeros of Hermite-Padé polynomials for a Nikishin system, Tr. Mat. Inst. Steklova 301 (2018), no. Kompleksnyĭ Analiz, Matematicheskaya Fizika i Prilozheniya, 259–275 (Russian, with Russian summary). English version published in Proc. Steklov Inst. Math. 301 (2018), no. 1, 245–261. MR 3841673, DOI 10.1134/S037196851802019X
  • Sergey P. Suetin, Hermite-Padé polynomials and analytic continuation: new approach and some results, arXiv:1806.08735, 2018.
  • S. P. Suetin, On the existence of a three-sheeted Nuttal surface in a certain class of infinite-valued analytic functions, Uspekhi Mat. Nauk 74 (2019), no. 2(446), 187–188 (Russian, with Russian summary); English transl., Russian Math. Surveys 74 (2019), no. 2, 363–365. MR 3951604, DOI 10.4213/rm9884
  • S. P. Suetin, On the equivalence of a scalar and a vector equilibrium problem for a pair of functions forming a Nikishin system, Mat. Zametki 106 (2019), no. 6, 904–916 (Russian, with Russian summary); English transl., Math. Notes 106 (2019), no. 5-6, 970–979. MR 4045676, DOI 10.4213/mzm12451
  • S. P. Suetin, Hermite-Padé polynomials and Shafer quadratic approximations for multivalued analytic functions, Uspekhi Mat. Nauk 75 (2020), no. 4(454), 213–214 (Russian); English transl., Russian Math. Surveys 75 (2020), no. 4, 788–790. MR 4153705, DOI 10.4213/rm9954
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Bibliographic Information
  • N. R. Ikonomov
  • Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
  • Email: nikonomov@math.bas.bg
  • S. P. Suetin
  • Affiliation: Steklov Mathematical Institute of Russian Academy of Sciences
  • MR Author ID: 190281
  • Email: suetin@mi-ras.ru
  • Published electronically: September 23, 2024
  • Additional Notes: The research of the second author was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 18-01-00764).
  • © Copyright 2024 N. R. Ikonomov and S. P. Suetin
  • Journal: Trans. Moscow Math. Soc. 2022, 33-54
  • MSC (2020): Primary 30F99, 41A21, 42C05
  • DOI: https://doi.org/10.1090/mosc/344