Kahane-Katznelson-de Leeuw theorem and absolute convergence of Fourier series

TL;DR

This work extends the Kahane-Katznelson-de Leeuw theorem to smoothness spaces on $\mathbb{T}^d$ by showing that for any $g \in W^{l,2}(\mathbb{T}^d)$ there exists $f \in C^l(\mathbb{T}^d)$ with $|\widehat{f}(n)| \ge |\widehat{g}(n)|$ and $\omega_r(D^l f,t)_\infty \approx \omega_r(D^l g,t)_2$ for all $t>0$, and provides an equivalent formulation for $r$-quasiconcave majorants. The authors then leverage this to solve Bernstein’s problem for the absolute convergence of multiple Fourier series, giving sharp criteria expressed through moduli of smoothness and Besov-type spaces, and they extend the analysis to absolute integrability of Fourier transforms on $\mathbb{R}^d$. Central to the approach are discretizing sequences, de la Vallée-Poussin-type constructions, and Littlewood-Paley-type decompositions that transfer control from Fourier coefficients to smoothness norms. The results yield precise, scale-aware conditions for membership in Wiener-type spaces $\mathcal{A}_p$ and illuminate the interplay between smoothness, spectral concentration, and absolute convergence in both periodic and non-periodic settings.

Abstract

We extend the Kahane-Katznelson-de Leeuw theorem to smoothness spaces by showing that for any $g \in W^{l,2}(\mathbb{T}^d)$, there exists a function $f\in C^l(\mathbb{T}^d)$ satisfying $|\widehat{f}(n)|\geq |\widehat{g}(n)|$ and $$ω_r(D^l f,t)_\infty \approx ω_r(D^l g,t)_2, \quad t>0. $$ We apply this result to solve the Bernstein problem of finding necessary and sufficient conditions for the absolute convergence of multiple Fourier series. Finally, we explore the absolute integrability of Fourier transforms.

Theorems & Definitions (23)

• Theorem 1.1: Kis24
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6: wainger65
• Theorem 1.7: Kis24
• Theorem 1.8
• Remark 1.9
• Lemma 2.1: Discretizing sequences, cf. gogatishvili2003discretization