The Laplace transform and polynomial approximation in L2

TL;DR

This paper establishes a sufficient condition based on the Laplace transform for the density of a measure to ensure polynomial density in L2 space, with implications for semiparametric model extensions.

Contribution

It introduces a new, simple criterion involving the Laplace transform for polynomial density in L2, and provides an alternative condition based on tail decay.

Findings

Laplace transform bounded near zero implies finite moments

Polynomials are dense in L2 under the new condition

An easier-to-verify tail decay condition is proposed

Abstract

This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by $L^2$. It is shown that if the Laplace transform of the measure in play is bounded in a neighbourhood of the origin, then the moments of all order are finite and the class of polynomials is dense in $L^2$. The existence of the moments of all orders is well known for the case where the measure is concentrated in the positive real line (see Feller, 1966), but the result concerning the polynomial approximation is original, even thought the proof is relatively simple. Additionally, an alternative stronger condition easier to be verified not involving the calculation of the Laplace transform is given. The condition essentially says that the density of the measure should have exponential decaying tails. The tools presented are of interest for constructing semiparametric extensions of classic parametric models.