# On some shift invariant integral operators, univariate case

George A. Anastassiou; Heinz H. Gonska

Annales Polonici Mathematici (1995)

- Volume: 61, Issue: 3, page 225-243
- ISSN: 0066-2216

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topGeorge A. Anastassiou, and Heinz H. Gonska. "On some shift invariant integral operators, univariate case." Annales Polonici Mathematici 61.3 (1995): 225-243. <http://eudml.org/doc/262320>.

@article{GeorgeA1995,

abstract = {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.},

author = {George A. Anastassiou, Heinz H. Gonska},

journal = {Annales Polonici Mathematici},

keywords = {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},

language = {eng},

number = {3},

pages = {225-243},

title = {On some shift invariant integral operators, univariate case},

url = {http://eudml.org/doc/262320},

volume = {61},

year = {1995},

}

TY - JOUR

AU - George A. Anastassiou

AU - Heinz H. Gonska

TI - On some shift invariant integral operators, univariate case

JO - Annales Polonici Mathematici

PY - 1995

VL - 61

IS - 3

SP - 225

EP - 243

AB - 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.

LA - eng

KW - 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

UR - http://eudml.org/doc/262320

ER -

## References

top- [1] G. Anastassiou, C. Cottin and H. Gonska, Global smoothness of approximating functions, Analysis 11 (1991), 43-57. Zbl0722.41021
- [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. Zbl0795.41011
- [3] G. Anastassiou and X. M. Yu, Monotone and probabilistic wavelet approximation, Stochastic Anal. Appl. 10 (1992), 251-264. Zbl0763.41011
- [4] G. Anastassiou and X. M. Yu, Convex and coconvex-probabilistic wavelet approximation, Stochastic Anal. Appl., 507-521. Zbl0804.41018

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