Laguerre polynomials and the inverse Laplace transform using discrete data

TL;DR

This paper introduces a method to approximate functions from their Laplace transform values using Laguerre polynomial expansions and Lagrange polynomial coefficients, addressing the ill-posedness of the inverse Laplace problem.

Contribution

It presents a novel approach combining Laguerre polynomial expansion and Lagrange interpolation to stabilize the inverse Laplace transform from discrete data.

Findings

Provides a stable approximation method for inverse Laplace transform

Demonstrates effectiveness with theoretical analysis

Addresses ill-posedness of the problem

Abstract

We consider the problem of finding a function defined on $(0,\infty)$ from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution.