Translation-based completeness on compact intervals

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