Lp Convergence with Rates of Smooth Poisson-Cauchy Type Singular Operators

TL;DR

This paper investigates the convergence rates of smooth Poisson-Cauchy type singular integral operators to the identity operator in Lp spaces, providing inequalities involving higher order smoothness measures.

Contribution

It extends the analysis of Poisson-Cauchy operators by establishing convergence rates in Lp norms using higher order smoothness and derivatives.

Findings

Derived inequalities involving higher order Lp modulus of smoothness.

Established convergence rates of operators to the identity in Lp norm.

Analyzed the role of higher order derivatives in convergence behavior.

Abstract

In this article we continue the study of smooth Poisson-Cauchy Type singular integral operators on the line regarding their convergence to the unit operator with rates in the Lp norm, p greater equal one. The related established inequalities involve the higher order Lp modulus of smoothness of the engaged function or its higher order derivative.