Rational approximation of the Mittag-Leffler functions

N. V. Ryabchenko ; A. P. Starovoitov

Problemy fiziki, matematiki i tehniki, no. 1 (2021), pp. 65-68.

Résumé

It is shown that for $m-1\le n$ the Padé approximants $\{\pi_{n,m}(\cdot;F_\gamma)\}$, which locally deliver the best rational approximations to the Mittag-Leffler functions $F_\gamma$, approximate the $F_\gamma$ as $n\to\infty$ uniformly on the compact set $D=\{z:|z|\le1\}$ at a rate asymptotically equal to the best possible one. In particular, analogues of the well-know results of Braess and Trefethen relating to the approximation of $\exp(z)$ are proved for the Mittag-Leffler functions.

Détail

Keywords: Padé approximations, asymptotic equality, Mittag–Leffler functions, rational approximations.

Bibliographie

[1] M.G. Mittag-Leffler, “Sur la nouvelle fonction $E(x)$”, C.R. Akad. Sci. Paris, 137 (1903), 554–558

[2] M.M. Dzhrbashyan, Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966

[3] A.P. Starovoitov, N.A. Starovoitova, “Approksimatsii Pade funktsii Mittag-Lefflera”, Matem. sb., 198:7 (2007), 109–122 | Zbl

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

[5] E.B. Saff, “The convergence of rational functions of best approximation to the exponential function. II”, Proc. Amer. Math. Soc., 32 (1972), 187–194 | DOI | MR | Zbl

[6] E.B. Saff, “On the degree of best rational approximation to the exponential function”, J. Approx. Theory, 9:2 (1973), 97–101 | DOI | MR | Zbl

[7] D.S. Lubinsky, “Padé tables of entire functions of very slow and smooth growth. II”, Constr. Approx., 4 (1988), 321–339 | DOI | MR | Zbl

[8] A.L. Levin, D.S. Lubinsky, “Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients”, Constr. Approx., 6 (1990), 257–286 | DOI | MR | Zbl

[9] L.L. Berezkina, V.N. Rusak, “O nailuchshikh ratsionalnykh approksimatsiyakh nekotorykh tselykh funktsii”, Vestsi AN BSSR. Ser. fiz.-matem. navuk, 1990, no. 4, 27–32

[10] V.N. Rusak, Ta Khong Kuang, “O nailuchshikh ratsionalnykh approksimatsiyakh nekotorykh tselykh funktsii”, DAN BSSR, 34:10 (1990), 868–871

[11] V.N. Rusak, A.P. Starovoitov, “Approksimatsii Pade dlya tselykh funktsii s regulyarno ubyvayuschimi koeffitsientami Teilora”, Matem. sbornik, 193:9 (2002), 63–92 | Zbl

[12] A.L. Levin, D.S. Lubinsky, “Best rational approximation of entire functions whose Maclaurin series coefficients decrease rapidly and smoothly”, Trans. Amer. Math. Soc., 293 (1986), 533–545 | DOI | MR | Zbl

[13] D. Braess, “On the degree of best rational approximation to the exponential function”, J. Approx. Theory, 40:4 (1984), 375–379 | DOI | MR | Zbl

[14] L.N. Trefethen, “The asymptotic accuracy of rational best approximations to $e^z$ on a disk”, J. Approx. Theory, 40:4 (1984), 380–384 | DOI | MR

[15] A.P. Starovoitov, “Approksimatsii Ermita-Pade funktsii Mittag-Lefflera”, Trudy MIAN, 301, 2018, 241–258 | Zbl

[16] H. Van Rossum, “Systems of orthogonal and quasi-orthogonal polynomials connected with the Padé table I, II and III”, K. Nederl. Akad. Wetensch., Ser. A, 58 (1955), 517–534 ; 675–682 | DOI | MR | Zbl

[17] A.I. Aptekarev, “Ob approksimatsiyakh Pade k naboru $\{_1F_1(1,c;\lambda_iz)\}_{i=1}^k$”, Vestn. MGU. Seriya 1. Matematika. Mekhanika, 1981, no. 2, 58–62

[18] V.K. Dzyadyk, “Ob asimptotike diagonalnykh approksimatsii Pade funktsii $\sin z$, $\cos z$, $\sh z$ i $\ch z$”, Matem. sbornik, 108 (150):2 (1979), 247–267 | Zbl