On Some Shift-Invariant Integral Operators, Multivariate Case

Abstract

In recent papers, the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Also very recently, they studied the same property over ℝ, along with other characteristics, for some particular family of general shift-invariant integral operators. Here a generalization to ℝd, d ≥ 1, is given and everything can be transferred there from the univariate case. Namely, a general positive linear multivariate integral type operator is introduced through a convolution-like iteration of another general positive linear multivariate operator with a multivariate scaling type function. For this sufficient conditions are given for shift invariance, global smoothness preservation and its sharpness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic functions. Additionally, four examples of general specialized multivariate operators are given fulfilling all the above properties; especially, the inequalities for global smoothness preservation are sharp. In this article global smoothness preservation and convergence to the unit with rates involve a naturally arising suitable multivariate modulus of continuity.