Global Smoothness and Approximation by Activated Smooth Singular Operators

In this work we continue with the study of smooth activated singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 p , by involving higher order moduli of smoothness.Also we treat their activated simultaneous approximation to the unit operator with rates involving the modulus of smoothness.The derived Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.We involve five different activation functions.