Completeness of uniformly discrete translates in $L^p(\mathbb{R})$

TL;DR

The paper proves that there exists a uniformly discrete sequence $\{\lambda_n\}$ with $\lambda_n = n + o(1)$ and a Schwartz function $f$ such that, for any $N$, the tail translates $\{f(x - \lambda_n)\}_{n>N}$ are complete in $L^p(\mathbb{R})$ for every $p>1$, and this extends to a broad class of Banach function spaces. The approach combines Landau’s theory of complete exponentials on unions of intervals with a weighted-exponential framework and a Baire-category argument to produce a $\varphi \in I_0(\mathbb{R})$ whose weighted exponentials are complete in $I_0(\mathbb{R})$. Through Fourier duality, the completeness transfers from weighted exponentials to translates, yielding completeness of $\{f(x - \lambda_n)\}$ in $L^p(\mathbb{R})$ and in spaces satisfying the OU18b hypotheses. The method emphasizes positive translates and robustness under removing finite translates, and it broadens the known regime for uniform discreteness to guarantee completeness in $p>1$ spaces.

Abstract

We construct a real sequence $\{λ_n\}_{n=1}^{\infty}$ satisfying $λ_n = n + o(1)$, and a Schwartz function $f$ on $\mathbb{R}$, such that for any $N$ the system of translates $\{f(x - λ_n)\}$, $n > N$, is complete in the space $L^p(\mathbb{R})$ for every $p>1$. The same system is also complete in a wider class of Banach function spaces on $\mathbb{R}$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 3.1: Lan64
• Corollary 3.2
• proof
• Definition 4.1
• Theorem 4.2
• Lemma 4.3
• proof
• Theorem 4.4
• proof : Proof of Theorem \ref{['thm:landaucomp1']}