On Fredholm's Integral Equations on the Real Line, Whose Kernels Are Linear in a Parameter

TL;DR

This paper investigates Fredholm integral equations on the real line with kernels linear in a parameter, proving convergence of polynomial series and providing explicit solutions using Fredholm-determinant methods.

Contribution

It establishes convergence of Fredholm series for kernels linear in a parameter and offers explicit solutions to related integral equations on the real line.

Findings

Series converge in various function spaces

Explicit solutions to integral equations are obtained

Method extends classical Fredholm theory to new kernel forms

Abstract

In this paper, we study an infinite system of Fredholm series of polynomials in $\lambda$, formed, in the classical way, for a continuous Hilbert-Schmidt kernel on $\mathbb{R}\times\mathbb{R}$ of the form $\boldsymbol{H}(s,t)-\lambda\boldsymbol{S}(s,t)$, where $\lambda$ is a complex parameter. We prove a convergence of these series in the complex plane with respect to sup-norms of various spaces of continuous functions vanishing at infinity. The convergence results enable us to solve explicitly an integral equation of the second kind in $L^2(\mathbb{R})$, whose kernel is of the above form, by mimicking the classical Fredholm-determinant method.