Global smoothness and uniform convergence of smooth Poisson-Cauchy type singular operators

In this article we introduce the smooth Poisson-Cauchy type singular integral operators over the real line. Here we study their simultaneous global smoothness preservation property with respect to the Lp norm, 1⩽p⩽∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.