Vol. 2, No. 4, 2020

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Trigonometric series with a given spectrum

Yves Meyer

Vol. 2 (2020), No. 4, 881–906
DOI: 10.2140/tunis.2020.2.881

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

To the memory of Salah Baouendi

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mean periodic functions, almost periodic functions, trigonometric sums
Mathematical Subject Classification 2010
Primary: 42A32
Secondary: 42B10
Received: 15 January 2019
Revised: 24 July 2019
Accepted: 7 August 2019
Published: 19 January 2020
Yves Meyer
CMLA, ENS-Cachan, CNRS, Université Paris-Saclay