Does the ratio of Laplace transforms of powers of a function identify the function?

TL;DR

This paper investigates whether the ratio of Laplace transforms of different powers of a function can uniquely determine the original function, with positive results under certain conditions, relevant to economic auction models.

Contribution

It establishes conditions under which the ratio of Laplace transforms of powers uniquely identifies the function, extending inverse Laplace transform results.

Findings

Unique determination of the function from the ratio when one power is zero

Affirmative answer under smoothness assumptions for general powers

Application to auction theory in economics

Abstract

We study the following question: if $f$ is a nonzero measurable function on $[0,\infty)$ and $m$ and $n$ distinct nonnegative integers, does the ratio $\widehat{f^n}/\widehat{f^m}$ of the Laplace transforms of the powers $f^n$ and $f^m$ of $f$ uniquely determine $f$? The answer is yes if one of $m, n$ is zero, by the inverse Laplace transform. Under some assumptions on the smoothness of $f$ we show that the answer in the general case is also affirmative. The question arose from a problem in economics, specifically in auction theory where $f$ is the cumulative distribution function of a certain random variable. This is also discussed in the paper.