×

On some shift invariant integral operators, univariate case. (English) Zbl 0840.41016

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.

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
PDFBibTeX XMLCite
Full Text: DOI