An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on $\mathbb R$

TL;DR

This paper provides an analytic and numerical study of weighted singular Cauchy integrals with exponential weights on the real line, focusing on their properties, bounds, and polynomial interpolation techniques.

Contribution

It introduces new analytical and numerical methods for weighted singular Cauchy integrals with exponential weights, including bounds and interpolation strategies.

Findings

Derived bounds for derivatives of functions of the second kind.

Developed polynomial interpolation methods at zeros of orthonormal polynomials.

Analyzed the behavior of integrals with exponential weights and smooth external fields.

Abstract

This paper concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights $w:=\exp(-Q)$ with finite moments and with smooth external fields $Q:\mathbb R\to [0,\infty)$, with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to $w^2$. We also study bounds for the first derivatives of a class of functions of the second kind for $w^2$.