Asymptotic behaviour of Jain operators

2 years ago

(original), Approximation Theory, not-at-ICTP, paper

Abstract

The topic of the present paper are certain approximation operators acting on the space of continous functions on \([0,+\infty)\) having polynomial growth. The operators which were defined by Jain in 1972 are based on a probability distribution which is called generalized Poisson distribution. As a main result we derive a complete asymptotic expansion for the sequence of these operators.

Authors

Ulrich Abel
Technische Hochschule Mittelhessen, Department MND, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

Approximation by positive operators; Rate of convergence ; Degree of approximation; Asymptotic approximations · Asymptotic expansions

Paper coordinates

U. Abel, O. Agratini, Asymptotic behaviour of Jain operators, Numer Algor 71 (2016), 553–565. https://doi.org/10.1007/s11075-015-0009-3

PDF

requires subscription: https://doi.org/10.1007/s11075-015-0009-3

About this paper

Journal

Publisher Name

DOI

https://doi.org/10.1007/s11075-015-0009-3

Print ISSN

1572-9265

Online ISSN

1017-1398

google scholar link

1. Abel, U.: On the asymptotic approximation with bivariate operators of Bleimann, Butzer, and Hahn. J. Approx. Theory 97, 181–198 (1999)
2. Abel, U., Berdysheva, E.E.: Complete asymptotic expansion for multivariate Bernstein-Durrmeyer operators with Jacobi weights. J. Approx. Theory 162, 201–220 (2010)
3. Abel, U., Butzer, P.L.: Complete asymptotic expansion for generalized Favard operators. Constr. Approx. 35, 73–88 (2012)
4. Abel, U., Ivan, M.: Complete asymptotic expansions for Altomare operators. Mediterr. J. Math. 10, 17–29 (2013)
5. Agratini, O.: Approximation properties of a class of linear operators. Math. Meth. Appl. Sci. 36, 2353–2358 (2013)
6. Catinas, T., Otrocol, D.: Iterates of multivariate Cheney-Sharma operators. J. Comput. Anal. Appl. 15(7), 1240–1246 (2013)
7. Consul, P.C., Jain, G.C.: A generalization of the Poisson distribution. Technometrics 15, 791–799 (1973)
8. Consul, P.C., Jain, G.C.: On some interesting properties of the generalized poisson distribution. Biometr. Z. 15, 495–500 (1973)
9. Comtet, L.: Advanced Combinatorics. D. Reidel Publishing Co., Dordrecht, Holland (1974)
10. Farcas, A.: An asymptotic formula for Jain’s operators. Stud. Univ. Babes-Bolyai Math. 57, 511–517 (2012)
11. Gupta, V., Agarwal, R.P., Verma, D.K.: Approximation for a new sequence of summation-integral type operators. Adv. Math. Sci. Appl. 23(1), 35–42 (2013)
12. Gupta, V., Agarwal, R.P.: Convergence Estimates in Approximation Theory. Springer (2014)
13. Jain, G.C.: Approximation of functions by a new class of linear operators. J. Aust. Math. Soc. 13(3), 271–276 (1972)
14. Jordan, C.: Calculus of Finite Differences. Chelsea, New York (1965)
15. Mirakyan, G.: Approximation des fonctions continues au moyen de polynomes de la forme e−nx mn k=0 Ck,nxk . Dokl. Akad. Nauk. SSSR 31, 201–205 (1941)
16. Phillips, R.S.: An inversion formula for semi-groups of linear operators. Ann. Math. (Ser. 2) 59, 352–356 (1954)
17. Szasz, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45, 239–245 (1950)
18. Sikkema, P.C.: On some linear positive operators. Indag. Math. 32, 327–337 (1970)
19. Sikkema, P.C.: On the asymptotic approximation with operators of Meyer-K¨onig and Zeller. Indag. Math. 32, 428–440 (1970)
20. Tarabie, S.: On Jain-Beta linear operators. Appl. Math. Inf. Sci. 6(2), 213–216 (2012)
21. Tuenter, H.J.H.: On the generalized Poisson distribution. Stat. Neerl. 54, 374–376 (2000)
22. Umar, S., Razi, Q.: Approximation of function by a generalized Sz´asz operators. Commun. Fac. Sci. de l’Universite d’Ankara, Serie A1: Mathematique 34, 45–52 (1985)