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 ℝn, 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 ℝn 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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REFERENCES
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).
G. Anastassiou and X. M. Yu, Monotone and probabilistic wavelet approximation, Stoch. Anal. Appl. 10, 251-264 (1992).
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Anastassiou, G.A., Gal, S.G. Some Shift-Invariant Integral Operators, Univariate Case, Revisited. Journal of Computational Analysis and Applications 1, 3–23 (1999). https://doi.org/10.1023/A:1022662402497
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DOI: https://doi.org/10.1023/A:1022662402497
- 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