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

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 , q2, gives f q 5f* q an equivalent of Littlewood.

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 , q2, on obtient l’inégalité de Littlewood f q 5f* q .

@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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {26},
     number = {2},
     year = {1976},
     pages = {29-34},
     doi = {10.5802/aif.612},
     zbl = {0318.42009},
     mrnumber = {53 \#11292},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1976__26_2_29_0}
}
Montgomery, Hugh L. A note on rearrangements of Fourier coefficients. Annales de l'Institut Fourier, Volume 26 (1976) no. 2, pp. 29-34. doi : 10.5802/aif.612. http://www.numdam.org/item/AIF_1976__26_2_29_0/

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