Generalized moment representations and Pade approximants of analytic two-variable
functions

L. O. Chernetska

The theorems on construction of rational approximation by means of the method of generalized moment representations are generalized. Under conditions of the generalized theorems the formulas for the errors of approximation are established. The two-dimensional Pade approximants for some analytic two-variable functions are constructed.

1. Golub A.P., Chernetska L.O. Two-dimensional generalized moment representations and rational approximants of two-variable functions// Ukr. Mat. Zh. – 2013. – V.65, №8. – P. 1035–1058. (in Ukrainian)

2. Golub A.P., Chernetska L.O. Two-dimensional generalized moment representations and Pade approximants for some Humbert series// Ukr. Mat. Zh. – 2013. – V.65, №10. – P. 1315–1331. (in Ukrainian)

3. Dzjadyk V.K. On a generalization of the moment problem// Dokl. Akad. Nauk Ukrain. – 1981. – V.6. – P. 8–12. (in Ukrainian)

4. Alabiso C., Butera P. $N$-variable rational approximants and method of moments// J. Math. Phys. – 1975. – V.16, №4. – P. 840–845.

5. Cuyt A. How well can the concept of Pade approximant be generalized to the multivariate case?// J. Comput. Appl. Math. – 1999. – V.105, №1,2. – P. 25–50.

6. Hughes Jones R. General rational approximants in $N$ variables// J. Approx. Theory. – 1976. – V.16. – P. 201 – 233.

7. Lutterodt C. A two-dimensional analogue of Padґe approximant theory// J. Phys. A.: Math. – 1974. – V.7. – P. 1027–1037.

8. Zhou P. Explicit construction for multivariate Padґe approximants// J. Comput. Appl. Math. – 1997. – V.79. – P. 1–17.

9. Cuyt A., Driver K., Tan J., Verdonk B. Exploring multivariate Padґe approximants for multiple hypergeometric series// Adv. Comput. Math. – 1999. – V.10, №1. – P. 29–49.

10. Cuyt A., Tan J., Zhou P. General order multivariate Padґe approximants for pseudo-multivariate functions// Math. Comput. – 2006. – V.75, №254. – P. 727–741.

11. Borwein P.B., Cuyt A., Zhou P. Explicit construction of general multivariate Pade approximants to an Appell function// Adv. Comput. Math. – 2005. – V.22, №3. – P. 249–273.

12. Baker G.A., Graves–Morris P.R. Pade approximants. – M.: Mir, 1986. – 502 p. (in Russian)

13. Rudin W. Functional analysis. – M.: Mir, 1975. – 444 p. (in Russian)

14. Bateman H., Erdelyi A. Higher transcendental functions. – M.: Nauka, 1973, V.1. – 296 p. (in Russian)

15. Kantorovich L.V., Akilov G.P. Functional analysis. – M.: Nauka, 1977. – 744 p. (in Russian)

16. Abramowitz M., Stegun I. Handbook of mathematical functions with formulas, graphs and mathematical tables. – M.: Nauka, 1979. – 832 p. (in Russian)

17. Karlin S., Studden W.J. Tchebycheff systems: with applications in analysis and statistics. – M.: Nauka, 1976. – 568 p. (in Russian)

	Generalized moment representations and Pade approximants of analytic two-variable functions(in Ukrainian)
Author	L. O. Chernetska chernets.liliya@yandex.ua Institute of Mathematics NASU
Abstract	The theorems on construction of rational approximation by means of the method of generalized moment representations are generalized. Under conditions of the generalized theorems the formulas for the errors of approximation are established. The two-dimensional Pade approximants for some analytic two-variable functions are constructed.
Keywords	Pade approximation; biorthogonal polynomial; Appell series
Reference	1. Golub A.P., Chernetska L.O. Two-dimensional generalized moment representations and rational approximants of two-variable functions// Ukr. Mat. Zh. – 2013. – V.65, №8. – P. 1035–1058. (in Ukrainian) 2. Golub A.P., Chernetska L.O. Two-dimensional generalized moment representations and Pade approximants for some Humbert series// Ukr. Mat. Zh. – 2013. – V.65, №10. – P. 1315–1331. (in Ukrainian) 3. Dzjadyk V.K. On a generalization of the moment problem// Dokl. Akad. Nauk Ukrain. – 1981. – V.6. – P. 8–12. (in Ukrainian) 4. Alabiso C., Butera P. $N$-variable rational approximants and method of moments// J. Math. Phys. – 1975. – V.16, №4. – P. 840–845. 5. Cuyt A. How well can the concept of Pade approximant be generalized to the multivariate case?// J. Comput. Appl. Math. – 1999. – V.105, №1,2. – P. 25–50. 6. Hughes Jones R. General rational approximants in $N$ variables// J. Approx. Theory. – 1976. – V.16. – P. 201 – 233. 7. Lutterodt C. A two-dimensional analogue of Padґe approximant theory// J. Phys. A.: Math. – 1974. – V.7. – P. 1027–1037. 8. Zhou P. Explicit construction for multivariate Padґe approximants// J. Comput. Appl. Math. – 1997. – V.79. – P. 1–17. 9. Cuyt A., Driver K., Tan J., Verdonk B. Exploring multivariate Padґe approximants for multiple hypergeometric series// Adv. Comput. Math. – 1999. – V.10, №1. – P. 29–49. 10. Cuyt A., Tan J., Zhou P. General order multivariate Padґe approximants for pseudo-multivariate functions// Math. Comput. – 2006. – V.75, №254. – P. 727–741. 11. Borwein P.B., Cuyt A., Zhou P. Explicit construction of general multivariate Pade approximants to an Appell function// Adv. Comput. Math. – 2005. – V.22, №3. – P. 249–273. 12. Baker G.A., Graves–Morris P.R. Pade approximants. – M.: Mir, 1986. – 502 p. (in Russian) 13. Rudin W. Functional analysis. – M.: Mir, 1975. – 444 p. (in Russian) 14. Bateman H., Erdelyi A. Higher transcendental functions. – M.: Nauka, 1973, V.1. – 296 p. (in Russian) 15. Kantorovich L.V., Akilov G.P. Functional analysis. – M.: Nauka, 1977. – 744 p. (in Russian) 16. Abramowitz M., Stegun I. Handbook of mathematical functions with formulas, graphs and mathematical tables. – M.: Nauka, 1979. – 832 p. (in Russian) 17. Karlin S., Studden W.J. Tchebycheff systems: with applications in analysis and statistics. – M.: Nauka, 1976. – 568 p. (in Russian)
Pages	201-213
Volume	41
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Generalized moment representations and Pade approximants of analytic two-variable functions(in Ukrainian)