Using wavelets for Szász-type operators

Authors

Keywords:

compactly supported wavelets, Szász type operators, second-order Lipschitz continuity, inverse theorems, K-functional
Abstract views: 257

Abstract

Szász-Mirakjan operators extend the classical Bernstein operators and are useful tools for approximating continuous functions on the infinite interval \([0, \infty)\).
These operators have integral variations of Kantorovich and Durrmeyer types in order to approximate \(L_p\) functions with \(1 \leq p <\infty\), but the integral operators cannot be used to characterize the second-order Lipschitz continuity of continuous functions.
In this paper we introduce a class of Szász type operators by means of Daubechies' compactly supported wavelets. These new operators can be used to characterize the second-order Lipschitz continuity of continuous functions and to approximate \(L_p\) functions.

We also provide direct and inverse theorems of these operators for these purposes.

Downloads

Download data is not yet available.

References

M. Becker, Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J. 27(1978), pp. 127-142.

H. Beres and G.G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J. 21 (1972), pp. 693-708.

I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), pp. 909-996.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Appl. Math. 61, SIAM Publ., Philadelphia, 1992.

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Series in Computational Mathematics vol. 9, Springer-Verlag, Berlin/Heidelberg/New York, 1987.,

Zhu-Rui Gou and Ding-Xuan Zhou, Approixmation theorems for modified Szász operators, Acta Sci. Mat. (Sgedeg) 56 (1992), pp. 311-321.

M. Heilmann, Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl. 5 (1989), pp. 105-127.

D. Mach and Ding-Xuan Zhou, Approximation theorems for a class of Bernstein-Durrmeyer operators, preprint, 1993.

S.M. Mazhar and V. Totik, Approximation by modified Szász operators, Acta Sci. Math. (Szeged) 49 (1985), pp. 257-269.

V. Totik, Uniform approximation by Sza'sz-Mirakjan type operators, Acta Math. Hungar. 41 (1983), pp. 291-307.

Mao-Dong Ye and Ding-Xuan Zhou, A class of operators by means of three-diagonal matrices, J. Approx. Theory 78 (1994), pp. 239-259.

Ding-Xuan Zhou, On a paper of Mazhar and Totik, J. Approx. theory. 72 (1993), pp. 290-300.

Ding-Xuan Zhou, Uniform approximation by some Dyrrmeyer operators, Approx. Theory Appl. 6(1990), pp. 87-100.

Ding-Xuan Zhou, On smoothness characterized by Bernstein type operators, J. Approx. theory, to appear.

Downloads

Published

1995-08-01

How to Cite

Gonska, H. H., & Zhou, D.-X. (1995). Using wavelets for Szász-type operators. Rev. Anal. Numér. Théor. Approx., 24(1), 131–145. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art14

Issue

Vol 24, Nos 1-2 (1995)

Section

Articles

License

Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.