Metric unconditionality and Fourier analysis

Stefan Neuwirth

ArticleOriginal scientific text

Title

Metric unconditionality and Fourier analysis

Authors ¹

Affiliations

Équipe d'Analyse, Université Paris VI, Boîte 186, 4, place Jussieu, F-75252 Paris Cedex 05, France

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} ()

and

C_{E} ()

of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces

p_{E} ()

, p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between

L^{p}_{E} ()

and

L^{p + 2}_{E} ()

. 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 \geq 1}

with

| \frac{n_{k + 1}}{n_{k}} | \to \infty

, then

C_{E} ()

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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Title

Metric unconditionality and Fourier analysis

Affiliations

Abstract

Bibliography