Translation-based completeness on compact intervals

Given a compact interval I subset of R, and a function f that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates { f (<middle dot> <middle dot> - lambda ) : lambda is an element of 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).