Anastassiou, George A.; Gonska, Heinz H. On some shift invariant integral operators, univariate case. (English) Zbl 0840.41016 Ann. Pol. Math. 61, No. 3, 225-243 (1995). Summary: In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is \(\mathbb{R}\). 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. Cited in 3 Documents MSC: 41A35 Approximation by operators (in particular, by integral operators) 41A36 Approximation by positive operators 41A55 Approximate quadratures 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A18 Iteration of real functions in one variable 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A25 Rate of convergence, degree of approximation 41A99 Approximations and expansions 60E05 Probability distributions: general theory Keywords:Jackson type inequalities; sharp inequalities; modulus of continuity; integral operators; shift invariant operators; convolution type operators; shape preserving operators; probabilistic distribution function; global smoothness preservation PDFBibTeX XMLCite \textit{G. A. Anastassiou} and \textit{H. H. Gonska}, Ann. Pol. Math. 61, No. 3, 225--243 (1995; Zbl 0840.41016) Full Text: DOI