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

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 $\mathbf{H}(s,t)-\lambda \mathbf{S}(s,t)$, 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.

English version of [Novitskii, I. M. On the convergence of polynomial Fredholm series. (Russian) Dal'nevost. Mat. Zh. 9 (2009), no. 1-2, 131-139]. 5 pages

References in corpus (1)

• Integral operators with infinitely smooth bi-Carleman kernels of Mercer type