An Eigenfunction Approach to Conversion of the Laplace Transform of Point Masses on the Real Line to the Fourier Domain

Abstract

Motivated by applications in magnetic resonance relaxometry, we consider the following problem: Given samples of a function $t\mapsto \sum_{k=1}^K A_k\exp(-t\lambda_k)$, where $K\ge 2$ is an integer, $A_k\in\mathbb{R}$, $\lambda_k>0$ for $k=1,\cdots, K$, determine $K$, $A_k$'s and $\lambda_k$'s. Unlike the case in which the $\lambda_k$'s are purely imaginary, this problem is notoriously ill-posed. Our goal is to show that this problem can be transformed into an equivalent one in which the $\lambda_k$'s are replaced by $i\lambda_k$. We show that this may be accomplished by approximation in terms of Hermite functions, and using the fact that these functions are eigenfunctions of the Fourier transform. We present a preliminary numerical exploration of parameter extraction from this formalism, including the effect of noise. We do not claim to have eliminated the inherent ill-posedness of the original problem, as reflected in the numerical results.