Abstract:
It is shown that for m⩽n the Padé approximants
{πn,m(⋅;Fγ)}, which locally deliver the best rational
approximations to the Mittag-Leffler functions Fγ, approximate
the Fγ as n→∞ uniformly on the compact set
D={z:|z|⩽1} at a rate asymptotically
equal to the best possible one. In particular, analogues of the well-known
results of Braess and Trefethen relating to the approximation of expz
are proved for the Mittag-Leffler functions.
Bibliography: 28 titles.
\Bibitem{StaSta07}
\by A.~P.~Starovoitov, N.~A.~Starovoitova
\paper Pad\'e approximants of the Mittag-Leffler functions
\jour Sb. Math.
\yr 2007
\vol 198
\issue 7
\pages 1011--1023
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This publication is cited in the following 30 articles:
Y. O. Afolabi, T. A. Biala, Ibrahim O. Sarumi, B. A. Wade, “Global-Padé Approximation of the Three-Parameter Mittag-Leffler Function: Generalized Derivation and Numerical Implementation Issues”, Commun. Appl. Math. Comput., 2025
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Aljowhara H. Honain, Khaled M. Furati, 2023 International Conference on Fractional Differentiation and Its Applications (ICFDA), 2023, 1
Tobias Danczul, Clemens Hofreither, Joachim Schöberl, “A unified rational Krylov method for elliptic and parabolic fractional diffusion problems”, Numerical Linear Algebra App, 30:5 (2023)
N. V. Ryabchenko, A. P. Starovoitov, “Ratsionalnaya approksimatsiya funktsii Mittag–Lefflera”, PFMT, 2021, no. 1(46), 65–68
N. V. Ryabchenko, “Trigonometricheskie approksimatsii Pade spetsialnykh funktsii”, PFMT, 2021, no. 2(47), 81–83
Hernandez-Balaguera E., “Numerical Approximations on the Transient Analysis of Bioelectric Phenomena At Long Time Scales Via the Mittag-Leffler Function”, Chaos Solitons Fractals, 145 (2021), 110768
Sarumi I.O., Furati Kh.M., Khaliq A.Q.M., “Highly Accurate Global Pade Approximations of Generalized Mittag-Leffler Function and Its Inverse”, J. Sci. Comput., 82:2 (2020), UNSP 46
Jeng S.W., Kilicman A., “Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods”, Symmetry-Basel, 12:6 (2020), 959
Jeng S.W., Kilicman A., “Series Expansion and Fourth-Order Global Pade Approximation For a Rough Heston Solution”, Mathematics, 8:11 (2020), 1968
Starovoitov A.P. Kechko E.P., “Asymptotics For Hermite-Pade Approximants Associated With the Mittag-Leffler Functions”, Lobachevskii J. Math., 41:11, SI (2020), 2295–2302
Taghavian H., “The Use of Partition Polynomial Series in Laplace Inversion of Composite Functions With Applications in Fractional Calculus”, Math. Meth. Appl. Sci., 42:7 (2019), 2169–2189
Gatheral J., Radoicic R., “Rational Approximation of the Rough Heston Solution”, Int. J. Theor. Appl. Financ., 22:3 (2019), 1950010
Iyiola O.S., Asante-Asamani E.O., Wade B.A., “a Real Distinct Poles Rational Approximation of Generalized Mittag-Leffler Functions and Their Inverses: Applications to Fractional Calculus”, J. Comput. Appl. Math., 330 (2018), 307–317
M. V. Sidortsov, A. A. Drapeza, A. P. Starovoitov, “Skorost skhodimosti kvadratichnykh approksimatsii Ermita–Pade vyrozhdennykh gipergeometricheskikh funktsii”, PFMT, 2018, no. 1(34), 71–78
A. P. Starovoitov, “Hermite–Padé approximants of the Mittag-Leffler functions”, Proc. Steklov Inst. Math., 301 (2018), 228–244
M. V. Sidortsov, A. A. Drapeza, A. P. Starovoitov, “Approksimatsii Ermita–Pade vyrozhdennykh gipergeometricheskikh funktsii”, PFMT, 2017, no. 2(31), 69–74
A. P. Starovoitov, “Asymptotics of Diagonal Hermite–Padé Polynomials for the Collection of Exponential Functions”, Math. Notes, 102:2 (2017), 277–288
A. P. Starovoitov, E. P. Kechko, “On Some Properties of Hermite–Padé Approximants to an Exponential System”, Proc. Steklov Inst. Math., 298 (2017), 317–333
Caibin Zeng, Yang Quan Chen, “Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse”, FCAA, 18:6 (2015), 1492
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