An Eigenfunction Approach to Conversion of the Laplace Transform of Point Masses on the Real Line to the Fourier Domain
Motivated by applications in magnetic resonance relaxometry, we consider the following problem: Given samples of a function $t\mapsto \sum_{k=1}^K A_k\exp(-tλ_k)$, where $K\ge 2$ is an integer, $A_k\in\mathbb{R}$, $λ_k>0$ for $k=1,\cdots, K$, determine $K$, $A_k$'s and $λ_k$'s. Unlike the case in which the $λ_k$'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 $λ_k$'s are replaced by $iλ_k$. 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.