Multilevel interpolations for the generalized Nikishin system on a tree graph

Lysov, V.

doi:10.1090/mosc/338

Multilevel interpolations for the generalized Nikishin system on a tree graph
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by V. G. Lysov;
Translated by: Kristian B. Kiradjiev

Trans. Moscow Math. Soc. 2022, 291-304

DOI: https://doi.org/10.1090/mosc/338

Published electronically: September 23, 2024

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Abstract:

We study a multilevel interpolation problem for a system of Markov functions defined by a tree graph. The normality of all indices has been proven. Asymptotic properties are studied in terms of the vector equilibrium problem for logarithmic potential.

References

A. Angelesco, Sur deux extensions des fractions continues algébriques, C. R. Acad. Sci. Paris 168 (1919), 262–265.
A. I. Aptekarev, Asymptotics of polynomials of simultaneous orthogonality in the Angelescu case, Mat. Sb. (N.S.) 136(178) (1988), no. 1, 56–84 (Russian); English transl., Math. USSR-Sb. 64 (1989), no. 1, 57–84. MR 945900, DOI 10.1070/SM1989v064n01ABEH003294
A. I. Aptekarev, Strong asymptotics of polynomials of simultaneous orthogonality for Nikishin systems, Mat. Sb. 190 (1999), no. 5, 3–44 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 5-6, 631–669. MR 1702555, DOI 10.1070/SM1999v190n05ABEH000401
A. I. Aptekarev, S. A. Denisov, and M. L. Yattselev, Discrete Schrödinger operator on a tree, Angelesco potentials, and their perturbations, Tr. Mat. Inst. Steklova 311 (2020), no. Analiz i Matematicheskaya Fizika, 5–13 (Russian, with Russian summary). English version published in Proc. Steklov Inst. Math. 311 (2020), no. 1, 1–9. MR 4223979, DOI 10.4213/tm4124
Alexander I. Aptekarev, Sergey A. Denisov, and Maxim L. Yattselev, Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials, Trans. Amer. Math. Soc. 373 (2020), no. 2, 875–917. MR 4068253, DOI 10.1090/tran/7959
Alexander I. Aptekarev, Sergey A. Denisov, and Maxim L. Yattselev, Jacobi matrices on trees generated by Angelesco systems: asymptotics of coefficients and essential spectrum, J. Spectr. Theory 11 (2021), no. 4, 1511–1597. MR 4349670, DOI 10.4171/jst/380
A. I. Aptekarev, V. Kalyagin, G. López Lagomasino, and I. A. Rocha, On the limit behavior of recurrence coefficients for multiple orthogonal polynomials, J. Approx. Theory 139 (2006), no. 1-2, 346–370. MR 2220045, DOI 10.1016/j.jat.2005.09.011
A. I. Aptekarev, V. A. Kalyagin, V. G. Lysov, and D. N. Toulyakov, Equilibrium of vector potentials and uniformization of the algebraic curves of genus 0, J. Comput. Appl. Math. 233 (2009), no. 3, 602–616. MR 2582993, DOI 10.1016/j.cam.2009.02.080
A. I. Aptekarev and A. B. È. Koĭèlaars, Hermite-Padé approximations and ensembles of multiple orthogonal polynomials, Uspekhi Mat. Nauk 66 (2011), no. 6(402), 123–190 (Russian, with Russian summary); English transl., Russian Math. Surveys 66 (2011), no. 6, 1133–1199. MR 2963452, DOI 10.1070/RM2011v066n06ABEH004771
A. I. Aptekarev, M. A. Lapik, and V. G. Lysov, Direct and inverse problems for vector logarithmic potentials with external fields, Anal. Math. Phys. 9 (2019), no. 3, 919–935. MR 4014848, DOI 10.1007/s13324-019-00297-8
A. I. Aptekarev, G. Lopes Lagomasino, and A. Martines-Finkel′shteĭn, On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials, Uspekhi Mat. Nauk 72 (2017), no. 3(435), 3–64 (Russian, with Russian summary); English transl., Russian Math. Surveys 72 (2017), no. 3, 389–449. MR 3662458, DOI 10.4213/rm9769
A. I. Aptekarev and V. G. Lysov, Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants, Mat. Sb. 201 (2010), no. 2, 29–78 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 1-2, 183–234. MR 2656323, DOI 10.1070/SM2010v201n02ABEH004070
A. I. Aptekarev, V. G. Lysov, and D. N. Tulyakov, Random matrices with an external source and the asymptotics of multiple orthogonal polynomials, Mat. Sb. 202 (2011), no. 2, 3–56 (Russian, with Russian summary); English transl., Sb. Math. 202 (2011), no. 1-2, 155–206. MR 2798785, DOI 10.1070/SM2011v202n02ABEH004142
A. I. Aptekarev and V. G. Lysov, Multilevel interpolation for Nikishin systems and boundedness of Jacobi matrices on binary trees, Uspekhi Mat. Nauk 76 (2021), no. 4(460), 179–180 (Russian); English transl., Russian Math. Surveys 76 (2021), no. 4, 726–728. MR 4295022, DOI 10.4213/rm10017
Walter Van Assche, Jeffrey S. Geronimo, and Arno B. J. Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, Special functions 2000: current perspective and future directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., vol. 30, Kluwer Acad. Publ., Dordrecht, 2001, pp. 23–59. MR 2006283, DOI 10.1007/978-94-010-0818-1_{2}
Jeffrey S. Geronimo, Plamen Iliev, and Walter Van Assche, Alpert multiwavelets and Legendre-Angelesco multiple orthogonal polynomials, SIAM J. Math. Anal. 49 (2017), no. 1, 626–645. MR 3612898, DOI 10.1137/16M1064465
Marjolein Leurs and Walter Van Assche, Jacobi-Angelesco multiple orthogonal polynomials on an $r$-star, Constr. Approx. 51 (2020), no. 2, 353–381. MR 4076114, DOI 10.1007/s00365-019-09457-2
Andrei Martínez-Finkelshtein and Walter Van Assche, What is…a multiple orthogonal polynomial?, Notices Amer. Math. Soc. 63 (2016), no. 9, 1029–1031. MR 3525716, DOI 10.1090/noti1430
Bernhard Beckermann, Valery Kalyagin, Ana C. Matos, and Franck Wielonsky, Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses, Constr. Approx. 37 (2013), no. 1, 101–134. MR 3010212, DOI 10.1007/s00365-012-9165-z
M. Bertola, M. Gekhtman, and J. Szmigielski, The Cauchy two-matrix model, Comm. Math. Phys. 287 (2009), no. 3, 983–1014. MR 2486670, DOI 10.1007/s00220-009-0739-y
M. Bertola, M. Gekhtman, and J. Szmigielski, Cauchy biorthogonal polynomials, J. Approx. Theory 162 (2010), no. 4, 832–867. MR 2606648, DOI 10.1016/j.jat.2009.09.008
M. Bertola, M. Gekhtman, and J. Szmigielski, Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model, J. Math. Phys. 54 (2013), no. 4, 043517, 25. MR 3088819, DOI 10.1063/1.4802455
Marco Bertola and Thomas Bothner, Universality conjecture and results for a model of several coupled positive-definite matrices, Comm. Math. Phys. 337 (2015), no. 3, 1077–1141. MR 3339172, DOI 10.1007/s00220-015-2327-7
A. I. Bogolyubskii and V. G. Lysov, Constructive solution of one vector equilibrium problem, Dokl. Math. 101 (2020), no. 2, 90–92. Translation of Dokl. Akad. Nauk 491 (2020), 15–18. MR 4431640, DOI 10.1134/s1064562420020064
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, 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
Juan G. Criado del Rey and Arno B. J. Kuijlaars, A vector equilibrium problem for symmetrically located point charges on a sphere, Constr. Approx. 55 (2022), no. 3, 775–827. MR 4434024, DOI 10.1007/s00365-022-09566-5
A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and perturbation theory (Rome, 1998) World Sci. Publ., River Edge, NJ, 1999, pp. 23–37. MR 1844104
Steven Delvaux and Arno B. J. Kuijlaars, A graph-based equilibrium problem for the limiting distribution of nonintersecting Brownian motions at low temperature, Constr. Approx. 32 (2010), no. 3, 467–512. MR 2726442, DOI 10.1007/s00365-010-9106-7
Sergey A. Denisov and Maxim L. Yattselev, Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality, Adv. Math. 396 (2022), Paper No. 108114, 79. MR 4370474, DOI 10.1016/j.aim.2021.108114
U. Fidalgo Prieto, A. López García, G. López Lagomasino, and V. N. Sorokin, Mixed type multiple orthogonal polynomials for two Nikishin systems, Constr. Approx. 32 (2010), no. 2, 255–306. MR 2677882, DOI 10.1007/s00365-009-9077-8
U. Fidalgo Prieto and G. López Lagomasino, Nikishin systems are perfect, Constr. Approx. 34 (2011), no. 3, 297–356. MR 2852293, DOI 10.1007/s00365-011-9139-6
U. Fidalgo, G. Lopez Lagomasino, and S. Medina Peralta, Asymptotic of Cauchy biorthogonal polynomials, Mediterr. J. Math. 17 (2020), no. 1, Paper No. 22, 30. MR 4042967, DOI 10.1007/s00009-019-1455-2
A. A. Gončar, The rate of rational approximation of certain analytic functions, Mat. Sb. (N.S.) 105(147) (1978), no. 2, 147–163, 287 (Russian). MR 489648
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
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 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
L. G. González Ricardo and G. López Lagomasino, Strong asymptotic of Cauchy biorthogonal polynomials and orthogonal polynomials with varying measure, Constr. Approx. 56 (2022), no. 3, 577–618. MR 4519591, DOI 10.1007/s00365-022-09580-7
L. G. González Ricardo, G. López Lagomasino, and S. Medina Peralta, Logarithmic asymptotic of multi-level Hermite-Padé polynomials, Integral Transforms Spec. Funct. 32 (2021), no. 5-8, 493–511. MR 4280696, DOI 10.1080/10652469.2020.1792899
L. G. González Ricardo, G. López Lagomasino, and S. Medina Peralta, On the convergence of multi-level Hermite-Padé approximants, Phys. D 440 (2022), Paper No. 133487, 11. MR 4467709, DOI 10.1016/j.physd.2022.133487
Adrien Hardy and Arno B. J. Kuijlaars, Weakly admissible vector equilibrium problems [republication of MR2914740], J. Approx. Theory 170 (2013), 44–58. MR 3044044, DOI 10.1016/j.jat.2012.03.015
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
V. A. Kaljagin, A class of polynomials determined by two orthogonality relations, Mat. Sb. (N.S.) 110(152) (1979), no. 4, 609–627 (Russian). MR 562212
A. V. Komlov, The polynomial Hermite-Padé $m$-system for meromorphic functions on a compact Riemann surface, Mat. Sb. 212 (2021), no. 12, 40–76 (Russian, with Russian summary); English transl., Sb. Math. 212 (2021), no. 12, 1694–1729. MR 4344414, DOI 10.4213/sm9577
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
Arno B. J. Kuijlaars, Multiple orthogonal polynomial ensembles, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., vol. 507, Amer. Math. Soc., Providence, RI, 2010, pp. 155–176. MR 2647568, DOI 10.1090/conm/507/09958
Arno B. J. Kuijlaars, A vector equilibrium problem for Muttalib-Borodin biorthogonal ensembles, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 065, 15. MR 3518864, DOI 10.3842/SIGMA.2016.065
A. B. J. Kuijlaars and P. Román, Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths, J. Approx. Theory 162 (2010), no. 11, 2048–2077. MR 2732921, DOI 10.1016/j.jat.2010.06.003
M. A. Lapik, On families of vector measures that are equilibrium measures in an external field, Mat. Sb. 206 (2015), no. 2, 41–56 (Russian, with Russian summary); English transl., Sb. Math. 206 (2015), no. 1-2, 211–224. MR 3354971, DOI 10.4213/sm8347
I. A. Lopatin, On a generalization of the scalar approach to the problem of distribution of zeros of Hermite–Padé polynomials for the Nikishin system, Mat. Sb. 213 (2022) (in press) (Russian).
G. López Lagomasino and W. Van Assche, Riemann-Hilbert analysis for a Nikishin system, Mat. Sb. 209(7) (2018), 106–138; English translation in Sb. Math. 209(7) (2018), 1019–1050.
G. López Lagomasino and S. Medina Peralta, On the convergence of type I Hermite-Padé approximants, Adv. Math. 273 (2015), 124–148. MR 3311759, DOI 10.1016/j.aim.2014.12.025
G. López Lagomasino, S. Medina Peralta, and J. Szmigielski, Mixed type Hermite-Padé approximation inspired by the Degasperis-Procesi equation, Adv. Math. 349 (2019), 813–838. MR 3943369, DOI 10.1016/j.aim.2019.04.024
V. G. Lysov, Strong asymptotics of Hermite-Padé approximants for a system of Stieltjes functions with Laguerre weight, Mat. Sb. 196 (2005), no. 12, 99–122 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 11-12, 1815–1840. MR 2225208, DOI 10.1070/SM2005v196n12ABEH003741
V. G. Lysov, Mixed type Hermite-Padé approximants for a Nikishin system, Tr. Mat. Inst. Steklova 311 (2020), no. Analiz i Matematicheskaya Fizika, 213–227 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 311 (2020), no. 1, 199–213. MR 4223990, DOI 10.4213/tm4146
V. G. Lysov and D. N. Tulyakov, On a vector potential-theory equilibrium problem with the Angelesco matrix, Tr. Mat. Inst. Steklova 298 (2017), no. Kompleksnyĭ Analiz i ego Prilozheniya, 185–215 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 298 (2017), no. 1, 170–200. MR 3725056, DOI 10.1134/S037196851703013X
V. Lysov and F. Wielonsky, Strong asymptotics for multiple Laguerre polynomials, Constr. Approx. 28 (2008), no. 1, 61–111. MR 2357986, DOI 10.1007/s00365-006-0648-1
K. Mahler, Perfect systems, Compositio Math. 19 (1968), 95–166 (1968). MR 239099
André Markoff, Deux démonstrations de la convergence de certaines fractions continues, Acta Math. 19 (1895), no. 1, 93–104 (French). MR 1554864, DOI 10.1007/BF02402872
Andrei Martínez-Finkelshtein and Guilherme L. F. Silva, Critical measures for vector energy: asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight, Adv. Math. 349 (2019), 246–315. MR 3939592, DOI 10.1016/j.aim.2019.04.010
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 and V. N. Sorokin, Ratsional′nye approksimatsii i ortogonal′nost′, “Nauka”, Moscow, 1988 (Russian). MR 953788
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
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
V. N. Sorokin, Cyclic graphs and Apéry’s theorem, Uspekhi Mat. Nauk 57 (2002), no. 3(345), 99–134 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 3, 535–571. MR 1918856, DOI 10.1070/RM2002v057n03ABEH000512
V. N. Sorokin, On Salikhov’s integral, Trans. Moscow Math. Soc. , posted on (2016), 107–126. MR 3643967, DOI 10.1090/mosc/254
V. N. Sorokin, Multipoint Hermite-Padé approximants for three beta functions, Trans. Moscow Math. Soc. 79 (2018), 117–134. MR 3881461, DOI 10.1090/mosc/276
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
V. N. Sorokin, A generalization of the discrete Rodrigues formula for Meixner polynomials, Mat. Sb. 213 (2022), no. 11, 79–101 (Russian, with Russian summary); English transl., Sb. Math. 213 (2022), no. 11, 1559–1581. MR 4582606, DOI 10.4213/sm9765
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
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
Maxim L. Yattselev, Strong asymptotics of Hermite-Padé approximants for Angelesco systems, Canad. J. Math. 68 (2016), no. 5, 1159–1200. MR 3536931, DOI 10.4153/CJM-2015-043-3

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Multilevel interpolations for the generalized Nikishin system on a tree graph
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Abstract: