Discrete Translates in Function Spaces

TL;DR

This paper constructs a Schwartz function whose translates form a spanning set in various function spaces, including all $L^p$ spaces for $p > 1$, under small perturbations of the integer lattice.

Contribution

It introduces a specific Schwartz function that ensures spanning properties for translated sets in multiple function spaces, even with small perturbations of the integer grid.

Findings

The constructed function's translates span $L^p(R)$ for all $p > 1$.

Spanning property holds under exponentially small perturbations of integers.

Results extend to more general function spaces with weaker norms.

Abstract

We construct a Schwartz function $\varphi$ such that for every exponentially small perturbation of integers $\Lambda$, the set of translates $\{\varphi(t-\lambda), \lambda\in\Lambda\}$ spans the space $L^p(R)$, for every $p > 1$. This result remains true for more general function spaces $X$, whose norm is "weaker" than $L^1$ (on bounded functions).