  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 $N$.
