Uniform convergence with rates of smooth Gauss-Weierstrass singular integral operators

摘要

In this article, we introduce and study the smooth Gauss-Weierstrass singular integral operators on the line of very general kind. We establish their convergence to the unit operator with rates. The estimates are mostly sharp and they are pointwise or uniform. The established inequalities involve the higher order modulus of smoothness. To prove optimality we use mainly the geometric moment theory method.