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arXiv · 1507.03334

On the summablility of truncated double Fourier series

Ahmed A. Abdelhakim
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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 $N$.

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Ahmed A. Abdelhakim. 2015-07-13. On the summablility of truncated double Fourier series. https://arxiv.org/abs/1507.03334

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