On the summablility of truncated double Fourier series

TL;DR

This paper investigates the bounds of truncated double Fourier series in Lebesgue spaces with mixed norms, providing comprehensive estimates for all exponent values and examples of matrices that maximize these bounds.

Contribution

It offers new estimates for the summability of truncated double Fourier series in Lebesgue spaces with mixed norms for all exponent ranges, including extremal matrix examples.

Findings

Derived bounds for all exponent values in Lebesgue spaces

Provided extremal matrix examples maximizing the bounds

Established independence of bounds from matrix size

Abstract

We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2\pi \imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite double sums of its coefficients. We obtain these estimates for all possible values of the exponents involved then we provide examples of matrices in ${\mathbb{C}}^{M\times N}$ that maximize some of them up to a constant independent of $M$ and $N$.