The Littlewood problem and non-harmonic Fourier series

Philippe Jaming, Karim Kellay, Chadi Saba

Abstract:

The aim of this note is to prove a lower bound of the $L^1$-norm
of non-harmonic trigonometric polynomials of the form
$$
C\sum_{k=1}^N\dfrac{|a_k|}{k}\leq \frac{1}{T}\int_{-T/2}^{T/2}|\sum_{k=1}^Na_ke^{2i\pi\lambda_kt}\,\mbox{d}t
$$
where $T>1$, $C=C(T)$ is an explicit constant that depends on $T$ only,
$\lambda_k$ are real numbers with $|\lambda_k-\lambda_\ell|\geq 1$ and $a_k$ are complex numbers.
This extends to the non-harmonic case the Littlewood conjecture as solved by McGehee, Pigno, Smith \cite{MPS} and Konyagin \cite{Kon} and was previously obtained by Nazarov \cite{Naz} with a non-explicit constant $C$.

Submission Type: Full Paper