Translation-based completeness on compact intervals

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.

arXiv comment10 pages, to appear in J. Approx. Theory

References in corpus (5)

• Injectivity of Gabor phase retrieval from lattice measurements
• Density in Approximation Theory
• Wiener's 'closure of translates' problem and Piatetski-Shapiro's uniqueness phenomenon
• Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions
• From completeness of discrete translates to phaseless sampling of the short-time Fourier transform