Some Shift-Invariant Integral Operators, Univariate Case, Revisited |
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Authors: | George A. Anastassiou Sorin G. Gal |
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Affiliation: | (1) Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, 38152;(2) Department of Mathematics, University of Oradea, Str. Armatei Romane 5, 3700 Oradea, Romania |
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Abstract: | In recent articles the first author and H. Gonska [e.g., see G. Anastassiou, C. Cottin, and H. Gonska, Global smoothness of approximating functions, Analysis, 11, 43–57 (1991); G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] studied global smoothness preservation by some univariate and multivariate linear operators over compact domains and n, n 1. In particular, they studied a very general positive linear integral type operator [e.g., see G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] over n that was introduced through a convolution-like integration of another general positive linear operator with a scaling-type function. In this article the authors, among others, extend and generalize [G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)]. Also certain new similar but more general integral operators are introduced and studied. These operators arise in a natural way, and for all these sufficient conditions are given for shift invariance, preservation of higher-order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preservation, and preservation of continuous probabilistic distribution functions. Several examples of very general specialized operators, old and new, are given that satisfy all the above properties. |
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Keywords: | Higher-order global smoothness preservation convergence to the unit with rates Jackson-type inequalities sharp inequalities first modulus of continuity higher-order modulus of smoothness integral operators shift-invariant operators convolution-type operators shape-preserving operators probabilistic distribution function scaling function |
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