METRIC UNCONDITIONALITY AND FOURIER-ANALYSIS

We investigate several aspects of almost 1-unconditionality. We charac terize the metric unconditional approximation property (umap) in terms of ''block unconditionality''. Then we focus on translation invariant subspaces L-E(p)(T) and C-E(T) of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work show s that the spaces L-E(p)(T), p an even integer, have a singular behavi our from the almost isometric point of view: property (umap) does not interpolate between L-E(p) (T) and L-E(p+2)(T). These arithmetical con ditions are used to construct counterexamples for several natural ques tions and to investigate the maximal density of such sets E. We also p rove that if E = {n(k)}k greater than or equal to 1 with \n(k+1)/n(k)\ --> infinity, then C-E(T) has (umap) and we get a sharp estimate of t he Sidon constant of Hadamard sets. Finally, we touch on the relations hip of metric unconditionality and probability theory.