Metric unconditionality and Fourier analysis

Stefan Neuwirth

Studia Mathematica (1998)

  • Volume: 131, Issue: 1, page 19-62
  • ISSN: 0039-3223

Abstract

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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 E p ( ) 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 E p ( ) and L E p + 2 ( ) . 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 ( ) 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.

How to cite

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Neuwirth, Stefan. "Metric unconditionality and Fourier analysis." Studia Mathematica 131.1 (1998): 19-62. <http://eudml.org/doc/216561>.

@article{Neuwirth1998,
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 $E = \{n_k\}_\{k≥1\}$ with $|n_\{k+1\}/n_k| → ∞$, 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.},
author = {Neuwirth, Stefan},
journal = {Studia Mathematica},
keywords = {Hadamard set; Sidon constant; almost 1-unconditionality; metric unconditional approximation property},
language = {eng},
number = {1},
pages = {19-62},
title = {Metric unconditionality and Fourier analysis},
url = {http://eudml.org/doc/216561},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Neuwirth, Stefan
TI - Metric unconditionality and Fourier analysis
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 1
SP - 19
EP - 62
AB - 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 $E = {n_k}_{k≥1}$ with $|n_{k+1}/n_k| → ∞$, 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.
LA - eng
KW - Hadamard set; Sidon constant; almost 1-unconditionality; metric unconditional approximation property
UR - http://eudml.org/doc/216561
ER -

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