On the summablility of truncated double Fourier series

Abstract

We estimate the truncated double trigonometric series ${\sum }_{n=0}^N{\sum }_{m=0}^M{a}_{mn}{e}^{2\pi ı\left(mx+ny\right)}$, $a_{mn}\in \mathbb{C}$, 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 $N$.