Metric unconditionality and Fourier analysis

Abstract

We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property UMAP in terms of ``block unconditionality''. Then we focus on translation invariant subspaces $L^p_E(T)$ and $C_E(T)$ of functions on the circle and express block unconditionality as arithmetical conditions on $E$. Our work shows that the spaces $L^p_E(T)$, $p$ an even integer, have a singular behaviour from the almost isometric point of view: property UMAP does not interpolate between spaces $L^p_E(T)$ and $L^{p+2}_E(T)$. These arithmetical conditions are used to construct counterexamples for several natural questions and to investigate the maximal density of such sets $E$. We also prove that if $E=\{n_k\}_{k\ge1}$ with $|n_{k+1}/n_k|\to\infty$, then $C_E(T)$ has UMAP and we get a sharp estimate of the Sidon constant of Hadamard sets. Finally, we investigate the relationship of metric unconditionality and probability theory.