Completeness of Integer Translates in Function Spaces on R

Abstract

We show that each of the Banach spaces C 0 (R) and L p (R), 2<p< , contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system in L 2 (R). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weighted L 2 spaces, and in the multivariate setting. We also prove some results in the opposite direction.

Spectral subspaces of $L^p$ for $p<1$

“…Theorem 6.13 shows that a small perturbation of the set of integers can affect its properties greatly. Other examples that exhibit such oddity and instability of the subset Z of R were known before (see, e.g., [6,23] and [24]). …”

Completeness in $L^1 (\mathbb R)$ of discrete translates

“…This result was established in [1], and another proof can be obtained from results of [7] (see also [3] for a particular case).…”

On Translation and Affine Systems Spanning L1(ℝ)

“…Interest in these questions was renewed some years ago by the result by Nikolskii and Atzmon-Olevskii [14,1] that Zgenerators exist in L p (R), p > 2 (while trivially they cannot exist in L p (R) for 1 ≤ p ≤ 2). Later, generators in L 2 (R) for almost integer translates were constructed by Olevskii and Olevskii-Ulanowskii [16,17].…”