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

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 $w^2$.