On the ${L}^{1}$ norm of exponential sums
Annales de l'Institut Fourier, Tome 30 (1980) no. 2, pp. 79-89.

La norme ${L}^{1}$ d’un polynôme trigonométrique ${\sum }_{1}^{N}{a}_{j}\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\phantom{\rule{0.166667em}{0ex}}\left({\mathrm{in}}_{j}x\right),$ $|{a}_{j}|\ge 1$, dépasse

 $C\left(\mathrm{log}\phantom{\rule{0.166667em}{0ex}}N\right)/\left(\mathrm{log}\phantom{\rule{0.166667em}{0ex}}\mathrm{log}\phantom{\rule{0.166667em}{0ex}}N{\right)}^{2}.$

The ${L}^{1}$ norm of a trigonometric polynomial with $N$ non zero coefficients of absolute value not less than 1 exceeds a fixed positive multiple of $C\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{log}\phantom{\rule{0.166667em}{0ex}}N\right)/\left(\mathrm{log}\phantom{\rule{0.166667em}{0ex}}\mathrm{log}\phantom{\rule{0.166667em}{0ex}}N{\right)}^{2}.$

@article{AIF_1980__30_2_79_0,
author = {Pichorides, S. K.},
title = {On the $L^1$ norm of exponential sums},
journal = {Annales de l'Institut Fourier},
pages = {79--89},
publisher = {Institut Fourier},
volume = {30},
number = {2},
year = {1980},
doi = {10.5802/aif.785},
zbl = {0432.42001},
mrnumber = {81j:10058},
language = {en},
url = {archive.numdam.org/item/AIF_1980__30_2_79_0/}
}
Pichorides, S. K. On the $L^1$ norm of exponential sums. Annales de l'Institut Fourier, Tome 30 (1980) no. 2, pp. 79-89. doi : 10.5802/aif.785. http://archive.numdam.org/item/AIF_1980__30_2_79_0/

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