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.
@article{bwmeta1.element.bwnjournal-article-apmv61z3p225bwm,
author = {George A. Anastassiou and Heinz H. Gonska},
title = {On some shift invariant integral operators, univariate case},
journal = {Annales Polonici Mathematici},
volume = {62},
year = {1995},
pages = {225-243},
zbl = {0840.41016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv61z3p225bwm}
}
George A. Anastassiou; Heinz H. Gonska. On some shift invariant integral operators, univariate case. Annales Polonici Mathematici, Tome 62 (1995) pp. 225-243. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv61z3p225bwm/
[000] [1] G. Anastassiou, C. Cottin and H. Gonska, Global smoothness of approximating functions, Analysis 11 (1991), 43-57. | Zbl 0722.41021
[001] [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. | Zbl 0795.41011
[002] [3] G. Anastassiou and X. M. Yu, Monotone and probabilistic wavelet approximation, Stochastic Anal. Appl. 10 (1992), 251-264. | Zbl 0763.41011
[003] [4] G. Anastassiou and X. M. Yu, Convex and coconvex-probabilistic wavelet approximation, Stochastic Anal. Appl., 507-521. | Zbl 0804.41018