Distribution of zeros of the Hermite--Pad\'e polynomials for a~system of three functions, and the Nuttall condenser

R. K. Kovacheva ; S. P. Suetin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 176-199.

Résumé

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite–Padé polynomials for a set of $m$ multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) $m$-sheeted Riemann surface possessing certain properties. In this paper, for $m=3$, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\mathfrak R_3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak f_1,\mathfrak f_2,\mathfrak f_3$ that are rational on the constructed Riemann surface and satisfy the independence condition $\det\bigl[\mathfrak f_k(z^{(j)})\bigr]\not\equiv0$. In the case of $m=3$, we refine the main theorem from Nuttall's paper of 1981. In particular, we show that in this case the complement $\overline{\mathbb C}\setminus B$ of the open (possibly, disconnected) set $B\subset\overline{\mathbb C}$ introduced in Nuttall's paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.

Bibliographie

[1] Aptekarev A.I., “Asimptotika approksimatsii Ermita–Pade dlya pary funktsii s tochkami vetvleniya”, DAN, 422:4 (2008), 443–445 | MR | Zbl

[2] Aptekarev A.I., Buslaev V.I., Martines-Finkelshtein A., Suetin S.P., “Approksimatsii Pade, nepreryvnye drobi i ortogonalnye mnogochleny”, UMN, 66:6 (2011), 37–122 | DOI | MR | Zbl

[3] Aptekarev A.I., Kalyagin V.A., Asimptotika kornya $n$-i stepeni iz polinomov sovmestnoi ortogonalnosti i algebraicheskie funktsii, Preprint No60, In-t prikl. mat. im. M.V. Keldysha, M., 1986

[4] Aptekarev A.I., Koielaars A.E., “Approksimatsii Ermita–Pade i ansambli sovmestno ortogonalnykh mnogochlenov”, UMN, 66:6 (2011), 123–190 | DOI | MR | Zbl

[5] Aptekarev A.I., Lysov V.G., “Sistemy markovskikh funktsii, generiruemye grafami, i asimptotika ikh approksimatsii Ermita–Pade”, Mat. sb., 201:2 (2010), 29–78 | DOI | MR | Zbl

[6] Aptekarev A.I., Lysov V.G., Tulyakov D.N., “Sluchainye matritsy s vneshnim istochnikom i asimptotika sovmestno ortogonalnykh mnogochlenov”, Mat. sb., 202:2 (2011), 3–56 | DOI | MR | Zbl

[7] Aptekarev A.I., Tulyakov D.N., Geometry of Hermite–Padé approximants for system of functions $\{f,f^2\}$ with three branch points, Preprint N 77, Keldysh Inst. Appl. Math., Moscow, 2012 http://www.keldysh.ru/papers/2012/prep2012_77.pdf

[8] Aptekarev A.I., Yattselev M.L., Padé approximants for functions with branch points—strong asymptotics of Nuttall–Stahl polynomials, E-print, 2011, arXiv: 1109.0332 [math.CA]

[9] Buslaev V.I., “O skhodimosti mnogotochechnykh approksimatsii Pade kusochno analiticheskikh funktsii”, Mat. sb., 204:2 (2013), 39–72 | DOI | MR | Zbl

[10] Buslaev V.I., Martines-Finkelshtein A., Suetin S.P., “Metod vnutrennikh variatsii i suschestvovanie $S$-kompaktov”, Tr. MIAN, 279, 2012, 31–58 | MR

[11] Chirka E.M., Rimanovy poverkhnosti, Lekts. kursy NOTs, 1, MIAN, M., 2006 | DOI | DOI

[12] Delvo S., Lopes A., Lopes Lagomasino G., “Ob odnom semeistve sistem Nikishina s periodicheskimi rekurrentnymi koeffitsientami”, Mat. sb., 204:1 (2013), 47–78 | DOI | MR

[13] Dumas S., Sur le développement des fonctions elliptiques en fractions continues, Thèse, Impr. Zürcher Furrer, Zürich, 1908

[14] Fidalgo Prieto U., López Lagomasino G., “Nikishin systems are perfect”, Constr. Approx., 34:3 (2011), 297–356 | DOI | MR | Zbl

[15] Forster O., Rimanovy poverkhnosti, Mir, M., 1980 | MR

[16] Gonchar A.A., “Ratsionalnye approksimatsii analiticheskikh funktsii”, Sovr. probl. matematiki, 1, 2003, 83–106 | DOI | MR

[17] Gonchar A.A., Rakhmanov E.A., “O skhodimosti sovmestnykh approksimatsii Pade dlya sistem funktsii markovskogo tipa”, Tr. MIAN, 157, 1981, 31–48 | MR | Zbl

[18] Gonchar A.A., Rakhmanov E.A., “Ravnovesnye raspredeleniya i skorost ratsionalnoi approksimatsii analiticheskikh funktsii”, Mat. sb., 134:3 (1987), 306–352 | MR | Zbl

[19] Gonchar A.A., Rakhmanov E.A., Sorokin V.N., “Ob approksimatsiyakh Ermita–Pade dlya sistem funktsii markovskogo tipa”, Mat. sb., 188:5 (1997), 33–58 | DOI | MR | Zbl

[20] Gonchar A.A., Rakhmanov E.A., Suetin S.P., “O skhodimosti approksimatsii Pade ortogonalnykh razlozhenii”, Tr. MIAN, 200, 1991, 136–146 | MR | Zbl

[21] Gonchar A.A., Rakhmanov E.A., Suetin S.P., “Approksimatsii Pade–Chebysheva dlya mnogoznachnykh analiticheskikh funktsii, variatsiya ravnovesnoi energii i $S$-svoistvo statsionarnykh kompaktov”, UMN, 66:6 (2011), 3–36 | DOI | MR | Zbl

[22] Komlov A.V., Suetin S.P., “Asimptoticheskaya formula dlya dvukhtochechnogo analoga polinomov Yakobi”, UMN, 68:4 (2013), 183–184 | DOI | MR | Zbl

[23] Lapik M.A., “O nositele ekstremalnoi mery v vektornoi zadache ravnovesiya”, Mat. sb., 197:8 (2006), 101–118 | DOI | MR | Zbl

[24] Lapik M.A., “Ravnovesnaya mera vo vneshnem pole dlya vektornogo potentsiala s matritsei vzaimodeistviya Nikishina”, UMN, 67:3 (2012), 179–180 | DOI | MR | Zbl

[25] Martines-Finkelshtein A., Rakhmanov E.A., Suetin S.P., “Variatsiya ravnovesnoi energii i $S$-svoistvo statsionarnogo kompakta”, Mat. sb., 202:12 (2011), 113–136 | DOI | MR

[26] Martínez-Finkelshtein A., Rakhmanov E.A., Suetin S.P., “Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall's work 25 years later”, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012, 165–193 | DOI | MR

[27] Nikishin E.M., “O sovmestnykh approksimatsiyakh Pade”, Mat. sb., 113:4 (1980), 499–519 | MR | Zbl

[28] Nikishin E.M., Sorokin V.N., Ratsionalnye approksimatsii i ortogonalnost, Nauka, M., 1988 | MR | Zbl

[29] Nuttall J., “Hermite–Padé approximants to functions meromorphic on a Riemann surface”, J. Approx. Theory, 32:3 (1981), 233–240 | DOI | MR | Zbl

[30] Nuttall J., “The asymptotic behavior of Hermite–Padé polynomials”, Circuits Syst. Signal Process., 1:3–4 (1982), 305–309 | DOI | MR | Zbl

[31] Nuttall J., “Asymptotics of diagonal Hermite–Padé polynomials”, J. Approx. Theory, 42:4 (1984), 299–386 | DOI | MR | Zbl

[32] Nuttall J., “Asymptotics of generalized Jacobi polynomials”, Constr. Approx., 2:1 (1986), 59–77 | DOI | MR | Zbl

[33] Nuttall J., Trojan G.M., “Asymptotics of Hermite–Padé polynomials for a set of functions with different branch points”, Constr. Approx., 3:1 (1987), 13–29 | DOI | MR | Zbl

[34] Rakhmanov E.A., “K asimptotike mnogochlenov Ermita–Pade dlya dvukh markovskikh funktsii”, Mat. sb., 202:1 (2011), 133–140 | DOI | MR | Zbl

[35] Rakhmanov E.A., “Orthogonal polynomials and $S$-curves”, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012, 195–239 | DOI | MR

[36] Rakhmanov E.A., Suetin S.P., “Asimptotika polinomov Ermita–Pade I roda dlya pary funktsii, obrazuyuschikh sistemu Nikishina”, UMN, 67:5 (2012), 177–178 | DOI | MR | Zbl

[37] Rakhmanov E.A., Suetin S.P., “Raspredelenie nulei polinomov Ermita–Pade dlya pary funktsii, obrazuyuschei sistemu Nikishina”, Mat. sb., 204:9 (2013), 115–160 | DOI | MR | Zbl

[38] Sorokin V.N., “O mnogochlenakh sovmestnoi ortogonalnosti dlya diskretnykh mer Meiksnera”, Mat. sb., 201:10 (2010), 137–160 | DOI | MR | Zbl

[39] Springer Dzh., Vvedenie v teoriyu rimanovykh poverkhnostei, Izd-vo inostr. lit., M., 1960 | MR

[40] Stahl H., “Diagonal Padé approximants to hyperelliptic functions”, Ann. Fac. Sci. Toulouse. Sér. 6. Math., 1996, Spec. Iss., 121–193 | DOI | MR | Zbl

[41] Stahl H., “The convergence of Padé approximants to functions with branch points”, J. Approx. Theory, 91:2 (1997), 139–204 | DOI | MR | Zbl

[42] Suetin S.P., “O ravnomernoi skhodimosti diagonalnykh approksimatsii Pade dlya giperellipticheskikh funktsii”, Mat. sb., 191:9 (2000), 81–114 | DOI | MR | Zbl

[43] Suetin S.P., “O skhodimosti chebyshevskikh nepreryvnykh drobei dlya ellipticheskikh funktsii”, Mat. sb., 194:12 (2003), 63–92 | DOI | MR | Zbl

[44] Suetin S., On the distribution of zeros of the Hermite–Padé polynomials for three algebraic functions $1$, $f$, $f^2$ and the global topology of the Stokes lines for some differential equations of the third order, E-print, 2013, arXiv: 1312.7105 [math.CV]