Trigonometric series with a given spectrum

Let $Λ \subset ℝ^{n}$ be a closed and discrete set. The vector space consisting of all trigonometric sums whose frequencies belong to $Λ$ is denoted by $T_{Λ}$ . Given an exponent $p \in [1, \infty]$ we say that $Λ$ is $p$ -coherent if there exist a compact set $K \subset ℝ^{n}$ and a continuous function $ω$ defined on $ℝ^{n}$ with values in $[1, \infty)$ such that for every $P \in T_{Λ}$ and every $y \in ℝ^{n}$ one has ${(\int_{| x - y | \leq 1} | P (x) |^{p} d x)}^{1 ∕ p} \leq ω (y) {(\int_{K} | P (x) |^{p} d x)}^{1 ∕ p}$ . Several properties of $p$ -coherent sets are proved in this essay.

To the memory of Salah Baouendi

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Keywords

mean periodic functions, almost periodic functions, trigonometric sums

Mathematical Subject Classification 2010

Primary: 42A32

Secondary: 42B10

Milestones

Received: 15 January 2019

Revised: 24 July 2019

Accepted: 7 August 2019

Published: 19 January 2020

Authors

Yves Meyer
CMLA, ENS-Cachan, CNRS, Université Paris-Saclay France

Vol. 2, No. 4, 2020

Yves Meyer

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors