Unconditional structures of translates for $L_p(R^d)$

Abstract

We prove that a sequence $(f_i{)}_{i=1}^{\infty }$ of translates of a fixed $f\in L_p(R)$ cannot be an unconditional basis of $L_p(R)$ for any $1\le p<\infty$. In contrast to this, for every $2<p<\infty$, $d\in N$ and unbounded sequence $({\lambda }_n{)}_{n\in N}\subset R^d$ we establish the existence of a function $f\in L_p(R^d)$ and sequence $(g_n^{\ast }{)}_{n\in N}\subset L_p^{\ast }(R^d)$ such that $(T_{{\lambda }_n}f,g_n^{\ast }{)}_{n\in N}$ forms an unconditional Schauder frame for $L_p(R^d)$. In particular, there exists a Schauder frame of integer translates for $L_p(R)$ if (and only if) $2<p<\infty$.

22 pages

References in corpus (1)

• Upper and lower estimates for schauder frames and atomic decompositions