Solving the Stieltjes Integral Equation in Explicit Form

TL;DR

This paper introduces a new explicit resolvent kernel to solve the Stieltjes integral equation under more general integrability conditions, expanding the class of solvable equations and revealing growth properties of solutions.

Contribution

It develops a novel explicit resolvent kernel for the Stieltjes integral equation under milder conditions than previously possible.

Findings

New explicit kernel enables solutions under broader conditions

Solutions exhibit interesting growth properties

Combining kernels can produce more effective non-convolution kernels

Abstract

Due to its convolution nature, the Stieltjes integral equation can be diagonalized by Mellin transform. Several explicit resolvent kernels were obtained over the years, all of convolution type. The conditions on the given function under which these convolution kernels are able to solve the equation, are rather restrictive. Purpose of this paper is to solve the Stieltjes integral equation - in explicit form - under more general conditions than has been done so far. In fact, we merely bestow upon the given function the same integrability as upon the unknown function. To solve the equation under this mild condition, we construct a new explicit resolvent kernel. For the solutions obtained, we derive interesting growth properties. The new kernel demonstrates that combining known convolution kernels may well lead to a non-convolution kernel that is more effective.