A note on rearrangements of Fourier coefficients

Montgomery, Hugh L.

Annales de l'Institut Fourier, Tome 26 (1976), p. 29-34 / Harvested from Numdam

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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.

Publié le : 1976-01-01

MR 53 #11292 | Zbl 0318.42009 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.612

@article{AIF_1976__26_2_29_0,
     author = {Montgomery, Hugh L.},
     title = {A note on rearrangements of Fourier coefficients},
     journal = {Annales de l'Institut Fourier},
     volume = {26},
     year = {1976},
     pages = {29-34},
     doi = {10.5802/aif.612},
     mrnumber = {53 \#11292},
     zbl = {0318.42009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1976__26_2_29_0}
}

Montgomery, Hugh L. A note on rearrangements of Fourier coefficients. Annales de l'Institut Fourier, Tome 26 (1976) pp. 29-34. doi : 10.5802/aif.612. http://gdmltest.u-ga.fr/item/AIF_1976__26_2_29_0/

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