Uniform Approximation with Rates by Multivariate Generalized Discrete Singular Operators

Here we establish the uniform approximation properties 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 give quantitatively the pointwise and uniform convergence of these operators to the unit operator by involving the multivariate higher order modulus of smoothness.