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 , q2, on obtient l’inégalité de Littlewood f q 5f* 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 , q2, gives f q 5f* 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},
     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/

[1] G. A. Bachelis, On the upper and lower majorant properties of Lp(G), Quart. J. Math. (Oxford), (2), 24 (1973), 119-128. | MR | Zbl

[2] A. Baernstein, II, Integral means, univalent functions and circular symmetrizations, Acta Math., 133 (1974), 139-169. | MR | Zbl

[3] G. H. Hardy and J. E. Littelwood, Notes on the theory of series (XIII) : Some new properties of Fourier constants, J. London Math. Soc., 6 (1931), 3-9. | JFM | Zbl

[4] G. H. Hardy and J. E. Littelwood, A new proof of a theorem on rearrangements, J. London math. Soc., 23 (1949), 163-168. | MR | Zbl

[5] F. R. Keogh, Some inequalities of Littlewood and a problem on rearrangements, J. London Math. Soc., 36 (1961), 362-376. | MR | Zbl

[6] J. E. Littlewood, On a theorem of Paley, J. London Math. Soc., 29 (1954), 387-395. | MR | Zbl

[7] J. E. Littlewood, On inequalities between f and f⋆, J. London Math. Soc., 35 (1960), 352-365. | MR | Zbl

[8] H. L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Springer-Verlag, Vol. 227, (1971), 187 pp. | MR | Zbl

[9] R. E. A. C. Paley, Some theorems on orthogonal functions, Studia Math., 3 (1931), 226-238. | EuDML | JFM | Zbl

[10] H. S. Shapiro, Majorant problems for Fourier coefficients, to appear. | Zbl

[11] A. Zygmund, Trigonometric series, Second Edition, Cambridge University Press, 1968.

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