Quantitative Uniform Approximation by Generalized Discrete Singular Operators

Here we study 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 establish quantitatively the pointwise and uniform convergences of these operators to the unit operator by involving the uniform higher modulus of smoothness of a uniformly continuous function. AMS 2010 Mathematics Subject Classi…cation: 26A15, 26D15, 41A17, 41A25. Key Words and Phrases: Discrete singular operator, modulus of smoothness, uniform convergence. 1 Introduction This article is motivated mainly by "Sur les multiplicateurs d’interpolation, J. Math. Pures Appl., IX, 23(1944), 219-247", where J. Favard in 1944 introduced the discrete version of Gauss-Weierstrass operator (Fnf) (x) = 1 p n 1 X = 1 f n exp n n x 2 ; (1) n 2 N; which has the property that (Fnf) (x) converges to f(x) pointwise for each x 2 R; and uniformly on any compact subinterval of R; for each continuous function f (f 2 C(R)) that ful…lls jf(t)j AeBt2 ; t 2 R; where A; B are positive constants.