General Theory of Global Smoothness Preservation by Singular Integrals, Univariate Case

In this article it is established that the well-known singular integrals of Picard, Poisson–Cauchy, Gauss–Weierstrass, and their Jackson-type generalizations fulfill 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 established 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.