Étude des coefficients de Fourier des fonctions de $L^p(G)$

Abstract

We study the decrease at infinity of the Fourier coefficients of 2π-periodic integrable functions. Let λ n be a lacunary sequence of integers: λ n+1 ≥3λ n : the associated k-lacunary sequence is defined to be the sequence μ N k of integers which can be written as ±λ n 1 ±λ n 2 ±⋯±λ n k , n 1 >n 2 >⋯>n k . It is shown that if ∫ 0 2π |f|( Log + |f|) k/2 dx is finite, then ∑ N |f ^(μ N k )| 2 is finite. If λ n satisfies a more restrictive condition, then for every p, 1<p≤2, it is shown that if ∫ 0 2π |f| p dx is finite, then ∑ k (p-1)∑ N |f ^(μ N k )| 2 is finite. These results are generalized to other groups besides R/2πZ, and to other situations. It is also shown that every k lacunary sequence which converges in a set of positive measure is the Fourier series of a square summable function.

Locations

• Annales de l’institut Fourier - View - PDF