Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Starovoitov A. P. Quadratic Hermite – Padé Approximants of Exponential Functions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 4, pp. 387-395. DOI: 10.18500/1816-9791-2014-14-4-387-395, EDN: TAAMHV

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.12.2014
Full text:
download
(downloads: 160)
Language: 
Russian
Heading: 
Mathematics
UDC: 
517.538.52+517.538.53
DOI: 
10.18500/1816-9791-2014-14-4-387-395
EDN: 
TAAMHV

Quadratic Hermite – Padé Approximants of Exponential Functions

Autors: 
Starovoitov Alexander Pavlovich, Gomel State University named after Francisk Skorina, Belarus
Abstract: 

The paper deals with extremal properties of diagonal quadratic Hermite – Pad’e approximants of type I for exponential system  {eλjz}2j =0 with arbitrary real λ0, λ1, λ2. Proved theorems complement known results of P. Borwein, F. Wielonsky.

Key words: 
Hermite – Pad’e approximants of type I
quadratic Hermite – Pad’e approximants
asymptotic equality
saddle-point method
References: 
  1. Hermite C. Sur la généralisation des fractions continues algébriques // Ann. Math. Pura. Appl. Ser. 2A. 1883. Vol. 21. P. 289–308.
  2. Hermite C. Sur la fonction exponentielle // C. R. Akad. Sci.(Paris). 1873. Vol. 77. P. 182–293.
  3. Mahler K. Perfect systems // Comp. Math. 1968. Vol. 19, № 2. P. 95–166.
  4. Mahler K. Zur Approximation der Exponentialfunktion und des Logarithmus // J. Reine Angew. Math. 1931. Vol. 166. P. 118–150.
  5. Padé H. Memoire sur les developpements en fractions continues de la fonctial exponential // Ann. École Norm. Sup. (Paris). 1899. Vol. 16, № 3. P. 394–426.
  6. Aptekarev A. I., Stahl H. Asymptotics of Hermite – Padé polynomials // Progress in Approximation Theory / eds. A. A. Gonchar, E. B. Saff. N.Y. ; Berlin : Springer-Verlag, 1992. P. 127–167.
  7. Mahler K. Applications of some formulas by Hermite to the approximation of exponentials and logarithms // Math. Ann. 1967. Vol. 168. P. 200–227.
  8. Chudnovsky G. V. Hermite – Padé approximations to exponential functions and elementary estimates of the measure of irrationality of ¼ // Lecture Notes in Math. Vol. 925. N. Y. ; Berlin : Springer-Verlag, 1982. P. 299–322.
  9. Borwein P. B. Quadratic Hermite – Padé approximation to the exponential function // Const. Approx. 1986. Vol. 62. P. 291–302.
  10. Wielonsky F. Asymptotics of Diagonal Hermite – Padé Approximants to ez // J. Approx. Theory. 1997. Vol. 90, № 2. P. 283–298.
  11. Trefethen L. N. The asymptotic accuracy of rational best approximations to ez on a disk // J. Approx. Theory. 1984. Vol. 40, № 4. P. 380–384.
  12. Braess D. On the conjecture of Meinardus on rational approximation of ex // J. Approx. Theory. 1984. Vol. 40, № 4. P. 375–379.
  13. Старовойтов А. П. Аппроксимации Эрмита – Паде для системы функций Миттаг –Леффлера // Проблемы физики, математики и техники. 2013. № 1(14). C. 81–87.
  14. Аптекарев А. И. О сходимости рациональных аппроксимаций к набору экспонент // Вестн. Моск. ун-та. Сер. 1, Математика. Механика. 1981. № 1. С. 68–74.
  15. Сидоров Ю. В., Федорюк М. В., Шабунин М. И. Лекции по теории функций комплексного переменного. М. : Наука, 1989. 477 с.
  16. Walsh J. L. Interpolaton and approximation by rational functions in the complex domain. Publ. by the Amer. Math. Soc., 1960. 508 p.
  17. Маркушевич А. И. Теория аналитических функций : в 2 т. Т. 1. М. : Наука, 1967. 486 с.
  18. Pólya G., Szegö G. Problems and Theorems in Analysis. Vol. 1. Berlin : Springer-Verlag, 1972. 419 p.
Received: 
15.06.2014
Accepted: 
03.11.2014
Published: 
01.12.2014
  • 847 reads