Translation-based completeness on compact intervals

TL;DR

This paper characterizes when translates of certain Gaussian-polynomial functions are complete in continuous functions on a compact interval, linking completeness to the divergence of the reciprocal series of translation points.

Contribution

It extends Zalik's theorem by providing a complete characterization of translation-based completeness for functions involving Gaussian and polynomial factors.

Findings

Completeness depends on divergence of reciprocal series of translation set.

Characterization for arithmetic progression translation sets.

Conditions for generator functions with fast decay.

Abstract

Given a compact interval $I \subseteq \mathbb{R}$, and a function $f$ that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates $\{ f(\cdot - \lambda) : \lambda \in \Lambda \}$ are complete in $C(I)$ if and only if the series of reciprocals of $\Lambda$ diverges. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299-308]. An additional characterization is obtained when $\Lambda$ is an arithmetic progression, and the generator $f$ constitutes a linear combination of translates of a function with sufficiently fast decay.