Asymptotics of the type II Hermite--Pad\'e approximation of exponential functions with complex multipliers in the exponent

M. V. Sidortsov ; N. A. Starovoitova ; A. P. Starovoitov

Problemy fiziki, matematiki i tehniki, no. 1 (2017), pp. 73-77.

Résumé

The asymptotic behavior of diagonal Hermite–Padé polynomials and diagonal Hermite–Padé approximations of type II for the system of exponentials $\{e^{\lambda_pz}\}_{p=0}^k$ in which $\lambda_0=0$, while the rest $\lambda_p$ are the roots of the equation $\xi^k=1$ is determined. The theorems complement known results of H. Padé, D. Braess, A. I. Aptekarev, H. Stahl, F. Wielonsky, W. Van Assche, A. B. J. Kuijlaars, A. P. Starovoitov, obtained for the case, where the $\{\lambda_p\}_{p=0}^k$ — different real numbers.

Détail

Keywords: Hermite integrals, Hermite–Padé polynomials, Hermite–Padé approximations, asymptotic equality.

Bibliographie

[1] J. P. Boyd, “Chebyshev expansion on intervals with branch points with application to the root of Kepler's equation: a Chebyshev–Hermite–Pade method”, J. of Comput. and Appl. Math., 223:2 (2009), 693–702 | DOI | MR | Zbl

[2] B. Beckermann, V. Kalyagin, Ana C. Matos, F. Wielonsky, How well does the Hermite–Pade approximation smooth the Gibbs phenomenon?, Math. Comput., 80:274 (2011), 931–958 | DOI | MR | Zbl

[3] V. N. Sorokin, “Tsiklicheskie grafy i teorema Aperi”, Uspekhi matem. nauk, 57:3 (2002), 99–134 | DOI | Zbl

[4] S. P. Suetin, “Approksimatsii Pade i effektivnoe analiticheskoe prodolzhenie stepennogo ryada”, Uspekhi matem. nauk, 57:1 (2002), 45–142 | DOI | Zbl

[5] S. P. Suetin, “Raspredelenie nulei polinomov Pade i analiticheskoe prodolzhenie”, Uspekhi matem. nauk, 70:5(425) (2015), 121–174 | DOI | Zbl

[6] P. M. Bleher, A. B. J. Kuijlaars, “Random matrices with external source and multiple orthogonal polynomials”, Int. Math. Res. Not., 3 (2004), 109–129 | DOI | MR | Zbl

[7] A. I. Aptekarev, P. M. Bleher, A. B. J. Kuijlaars, “Large $n$ limit of Gaussian random matrices with external source, Part II”, Comm. Math. Phys., 259:2 (2005), 367–389 | DOI | MR | Zbl

[8] A. I. Aptekarev, V. G. Lysov, D. N. Tulyakov, “Globalnyi rezhim raspredeleniya sobstvennykh znachenii sluchainykh matrits s angarmonicheskim potentsialom i vneshnim istochnikom”, TMF, 159:1 (2009), 34–57 | DOI | Zbl

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

[10] V. A. Kalyagin, “Approksimatsii Ermita–Pade i spektralnyi analiz nesimmetrichnykh operatorov”, Matem. sbornik, 185:6 (1994), 79–100

[11] A. I. Aptekarev, V. A. Kalyagin, E. B. Saff, “Higher-order three-term recurrences and asymptotics of multiple orthogonal polynomials”, Constr. Approx., 30:2 (2009), 175–223 | DOI | MR | Zbl

[12] G. V. Chudnovsky, Hermite–Padé approximations to exponential functions and elementary estimates of the measure of irrationality of $\pi$, Lecture Notes in Math., Springer-Verlag, New York–Berlin, 1982 | DOI | MR

[13] W. Van Assche, “Continued fractions: from analytic number theory to constructive approximation”, Contemp. Math., 236, Amer. Math. Soc., 1999, 325–342 | DOI | Zbl

[14] K. Mahler, “Applications of some formulas by Hermite to the approximation of exponentials and logarithms”, Math. Ann., 168 (1967), 200–227 | DOI | MR | Zbl

[15] A. I. Aptekarev (red.), Ratsionalnye priblizheniya postoyannoi Eilera i rekurrentnye sootnosheniya, Sbornik statei, Sovr. probl. matem., 9, MIAN, M., 2007

[16] A. I. Aptekarev, H. Stahl, “Asymptotics of Hermite–Pade polynomials”, Progress in Approximation Theory, eds. A. A. Gonchar, E. B. Saff, Springer-Verlag, New York–Berlin, 1992 | MR | Zbl

[17] A. I. Aptekarev, V. I. Buslaev, A. Martines-Finkelshtein, S. P. Suetin, “Approksimatsii Pade, nepreryvnye drobi i ortogonalnye mnogochleny”, Uspekhi matem. nauk, 66:6(402) (2011), 37–122 | DOI | Zbl

[18] A. I. Aptekarev, A. E. Koielaars, “Approksimatsii Ermita–Pade i ansambli sovmestno ortogonalnykh mnogochlenov”, Uspekhi matem. nauk, 66:6(402) (2011), 123–190 | DOI | Zbl

[19] H. Stahl, “Asymptotics for quadratic Hermite–Padé polynomals associated with the exponential function”, Electronic Trans. Num. Anal., 14 (2002), 193–220 | MR

[20] E. M. Nikishin, V. N. Sorokin, Ratsionalnye approksimatsii i ortogonalnost, Nauka, M., 1988

[21] Dzh. Beiker ml., P. Greivs-Morris, Approksimatsii Pade, v. 1, Osnovy teorii; т. 2, Обобщения и приложения, Мир, М., 1986 | MR

[22] C. Hermite, “Sur la fonction exponentielle”, C.R. Akad. Sci. (Paris), 77 (1873), 18–293

[23] A. I. Aptekarev, “O skhodimosti ratsionalnykh approksimatsii k naboru eksponent”, Vestn. MGU. Ser. 1. Matematika. Mekhanika, 1981, no. 1, 68–74

[24] O. Perron, Die Lehre von den Kettenbruchen, Teubner, Leipzig–Berlin, 1929 | MR | Zbl

[25] A. B. J. Kuijlaars, H. Stahl, W. Van Assche, F. Wielonsky, “Type II Hermite–Padé approximation to the exponential function”, J. of Comput. and Appl. Math., 207 (2007), 227–244 | DOI | MR | Zbl

[26] A. P. Starovoitov, “Asimptotika approksimatsii Ermita–Pade sistemy eksponent”, Doklady Natsionalnoi akademii nauk Belarusi, 57:2 (2013), 5–10 | Zbl

[27] A. P. Starovoitov, “O svoistvakh approksimatsii Ermita–Pade dlya sistemy funktsii Mittag–Lefflera”, Doklady Natsionalnoi akademii nauk Belarusi, 57:1 (2013), 5–10 | Zbl

[28] A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag–Lefflera”, Problemy fiziki, matematiki i tekhniki, 2013, no. 1(14), 81–87

[29] A. P. Starovoitov, “Ob asimptotike approksimatsii Ermita–Pade dlya sistemy funktsii Mittag–Lefflera”, Izvestiya vuzov. Matematika, 2014, no. 9, 59–68

[30] A. P. Starovoitov, “Ermitovskaya approksimatsiya dvukh eksponent”, Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya Matematika. Mekhanika. Informatika, 13:1(2) (2013), 88–91 | Zbl

[31] A. V. Astafeva, A. P. Starovoitov, “Asimptoticheskie svoistva mnogochlenov Ermita”, Doklady Natsionalnoi akademii nauk Belarusi, 59:3 (2015), 5–11

[32] A. V. Astafeva, A. P. Starovoitov, “Approksimatsii Ermita–Pade eksponentsialnykh funktsii”, Matem. sbornik, 207:6 (2016), 3–26 | DOI

[33] D. Braess, “On the conjecture of Meinardus on rational approximation of $e^x$, II”, J. Approx. Theory, 40:4 (1984), 375–379 | DOI | MR | Zbl

[34] Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin, Lektsii po teorii funktsii kompleksnogo peremennogo, Nauka, M., 1989 | MR