Merve Kester

Lp approximation with rates by multivariate generalized discrete singular operators

Here we give the approximation properties with rates of multivariate generalized discrete versions of Picard, Gauss-Weierstrass, and Poisson-Cauchy singular operators over R N , N ? 1. We treat both the unitary and non-unitary cases of the operators above. We derive quantitatively Lp convergence of these operators to the unit operator by involving the Lp higher modulus of smoothness of an Lp function.

Voronovskaya Type Asymptotic Expansions for Multivariate Generalized Discrete Singular Operators

Here we give asymptotic expansions including simultaneous ones, for the multivariate generalized discrete versions of unitary Picard, Gauss-Weierstrass, and Poisson-Cauchy type singular operators. These are Voronovskaya type expansions and they are connected to the approximation properties of these operators.

Voronovskaya Like Asymptotic Expansions for Generalized Discrete Singular Operators

GENERALIZED DISCRETE SINGULAR OPERATORS MERVE KESTER* (MKESTER@MEMPHIS.EDU) DEPARTMENT OF MATHEMATICAL SCIENCES ABSTRACT Here we give Voronovskaya type asymptotic expansions for the generalized discrete versions of unitary Picard, Gauss-Weierstrass, and PoissonCauchy singular operators.This is a joint work with Dr. George A. Anastassiou. PROPOSITION 1 We assume that the operators P ∗ r,n,ξ (f ;x), W ∗ r,n,ξ (f ;x), and Q ∗ r,n,ξ (f ;x) ∈ R, for x ∈ R. This is the case when ‖f‖∞,R <∞. For k = 1, ..., n, we need