On some shift invariant integral operators, univariate case

• Autorzy: George A. Anastassiou , Heinz H. Gonska
• Języki publikacji: EN

Abstrakty

EN

In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic functions. Finally, four examples of very general specialized operators are presented fulfilling all the above properties; in particular, the inequalities for global smoothness preservation are proven to be sharp.

Słowa kluczowe

EN

global smoothness preservation; convergence to the unit with rates; Jackson type inequalities; sharp inequalities; modulus of continuity; integral operators; shift invariant operators; convolution type operators; shape preserving operators; probabilistic distribution function

Kategorie tematyczne

• 41A55: Approximate quadratures
• 26A18: Iteration
• 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)
• 41A36: Approximation by positive operators
• 41A99: None of the above, but in this section
• 60E05: Distributions: general theory
• 41A35: Approximation by operators (in particular, by integral operators)
• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski{\u\i}-type inequalities)
• 41A25: Rate of convergence, degree of approximation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1995
• Tom: 61
• Numer: 3
• Strony: 225-243

Daty

wydano

1995

otrzymano

1993-04-20

poprawiono

1994-10-15

Twórcy

autor

George A. Anastassiou

• Department of Mathematics, University of Duisburg, D-47048 Duisburg, Germany

autor

Heinz H. Gonska

• Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152, U.S.A.

Bibliografia

• [1] G. Anastassiou, C. Cottin and H. Gonska, Global smoothness of approximating functions, Analysis 11 (1991), 43-57.
• [2] G. Anastassiou, C. Cottin and H. Gonska, Global smoothness preservation by multivariate approximation operators, in: Israel Mathematical Conference Proc. 4, Weizmann Science Press, 1991, 31-44.
• [3] G. Anastassiou and X. M. Yu, Monotone and probabilistic wavelet approximation, Stochastic Anal. Appl. 10 (1992), 251-264.
• [4] G. Anastassiou and X. M. Yu, Convex and coconvex-probabilistic wavelet approximation, Stochastic Anal. Appl., 507-521.