General Theory of Global Smoothness Preservation by Univariate Singular Operators

In this chapter we show that the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson type generalizations satisfy the “global smoothness preservation” property. I.e., they “ripple” less than the function they are applied on, that is producing a nice and fit approximation to the unit. The related results are given over various spaces of functions and the associated inequalities involve different types of corresponding moduli of smoothness. Several times these inequalities are proved to be sharp, namely they are attained. Here we follow the basic study done by both authors [26].