A note on rearrangements of Fourier coefficients

Montgomery, Hugh L.

doi:10.5802/aif.612

Montgomery, Hugh L.

Annales de l'Institut Fourier, Tome 26 (1976) no. 2, pp. 29-34.

Résumé
Abstract

Soit $f (x) \sim Σ a_{n} e^{2 π i n x}, f * (x) \sim \sum_{n = 0}^{\infty} a *_{n} \cos 2 π n x$ , où la suite $a *_{n}$ est le réarrangement décroissant de la suite $| a_{n} |$ . Pour toute fonction $ψ$ positive, convexe et croissante, on a $∥ ψ (| f |^{2} ∥_{1} \leq ∥ ψ (20 | f * |^{2} ∥_{1}$ . Dans le cas particulier $ψ (t) = t^{q / 2}$ , $q \geq 2$ , on obtient l’inégalité de Littlewood $∥ f ∥_{q} \leq 5 ∥ f * ∥_{q}$ .

Let $f (x) \sim Σ a_{n} e^{2 π i n x}, f * (x) \sim \sum_{n = 0}^{\infty} a *_{n} \cos 2 π n x$ , where the $a *_{n}$ are the numbers $| a_{n} |$ rearranged so that $a_{n}^{*} ↘ 0$ . Then for any convex increasing $ψ$ , $∥ ψ (| f |^{2} ∥_{1} \leq ∥ ψ (20 | f * |^{2} ∥_{1}$ . The special case $ψ (t) = t^{q / 2}$ , $q \geq 2$ , gives $∥ f ∥_{q} \leq 5 ∥ f * ∥_{q}$ an equivalent of Littlewood.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.612

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     author = {Montgomery, Hugh L.},
     title = {A note on rearrangements of {Fourier} coefficients},
     journal = {Annales de l'Institut Fourier},
     pages = {29--34},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {26},
     number = {2},
     year = {1976},
     doi = {10.5802/aif.612},
     mrnumber = {53 #11292},
     zbl = {0318.42009},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.612/}
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Montgomery, Hugh L. A note on rearrangements of Fourier coefficients. Annales de l'Institut Fourier, Tome 26 (1976) no. 2, pp. 29-34. doi : 10.5802/aif.612. http://www.numdam.org/articles/10.5802/aif.612/

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