Using wavelets for Szász-type operators

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Keywords:

compactly supported wavelets, Szász type operators, second-order Lipschitz continuity, inverse theorems, K-functional
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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.

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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.

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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

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Vol 24, Nos 1-2 (1995)

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