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

Abstract

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.

Keywords:

• best constant
• Gauss–Weierstrass singular integral
• modulus of smoothness
• sharp inequality

AMS Subject Classifications:

• Primary: 26A15
• 41A17
• 41A35
• Secondary: 26D15
• 41A44