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1998 | 131 | 1 | 19-62
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Metric unconditionality and Fourier analysis

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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 $p_{E}(𝕋)$, p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L^{p}_{E}(𝕋)$ 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 $E = {n_k}_{k≥1}$ with $|n_{k+1}/n_k| → ∞$, then $C_{E}(𝕋)$ has umap and we get a sharp estimate of the Sidon constant of Hadamard sets. Finally, we touch on the relationship of metric unconditionality and probability theory.
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  • Équipe d'Analyse, Université Paris VI, Boîte 186, 4, place Jussieu, F-75252 Paris Cedex 05, France, neuwirth@ccr.jussieu.fr
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Bibliografia
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