B-splines approximate the Gaussian density and Hermite functions in Schwartz seminorms

TL;DR

This paper demonstrates that B-splines can approximate Gaussian densities and Hermite functions in Schwartz seminorms, establishing new links between spline theory and classical orthogonal polynomials.

Contribution

It proves B-splines approximate Gaussian densities in Schwartz seminorms and derives asymptotic formulas connecting Hermite and Laguerre polynomials.

Findings

B-splines tend to the Gaussian density in Schwartz seminorms

Asymptotic formulas connect Hermite and Laguerre polynomials

B-splines relate to the volumes of sections of the standard simplex

Abstract

We prove that B-splines with knots satisfying assumptions of the Berry-Esseen Theorem, which correspond directly to the volumes of sections of the standard simplex, tend to the Gaussian density in any Schwartz seminorm. As a consequence, we derive some asymptotic formulas connecting Hermite and Laguerre polynomials.