no. 2
A note on rearrangements of Fourier coefficients
Annales de l'Institut Fourier, Tome 26 (1976) no. 2, pp. 29-34.

Soit f(x)∼Σa n e 2πinx ,f*(x)∼∑ n=0 ∞ a* n cos 2πnx, 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 ≤∥ψ(20|f*| 2 ∥ 1 . Dans le cas particulier ψ(t)=t q/2 , q≥2, on obtient l’inégalité de Littlewood ∥f∥ q ≤5∥f*∥ q .

Let f(x)∼Σa n e 2πinx ,f*(x)∼∑ n=0 ∞ a* n cos 2πnx, 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 ≤∥ψ(20|f*| 2 ∥ 1 . The special case ψ(t)=t q/2 , q≥2, gives ∥f∥ q ≤5∥f*∥ q an equivalent of Littlewood.

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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},
     zbl = {0318.42009},
     mrnumber = {53 #11292},
     language = {en},
     url = {https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.612/

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