Approximation with rates by generalized discrete singular operators

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