High Order Approximation with Multivariate Generalized Trigonometric Type Singular Integral Operators


This research and survey chapter deals exclusively with the chapter of the approximation of generalized multivariate Trigonometric type singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. These operators are not in general positive linear operators. In particular we study the rate of convergence of these integral operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these integral operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of approximation.

Publication Title

Studies in Systems, Decision and Control