# A note on rearrangements of Fourier coefficients

• Volume: 26, Issue: 2, page 29-34
• ISSN: 0373-0956

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## Abstract

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Let $f\left(x\right)\sim \Sigma {a}_{n}{e}^{2\pi inx},f*\left(x\right)\sim {\sum }_{n=0}^{\infty }a{*}_{n}\phantom{\rule{0.166667em}{0ex}}\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}2\pi nx$, where the $a{*}_{n}$ are the numbers $|{a}_{n}|$ rearranged so that ${a}_{n}^{*}↘0$. Then for any convex increasing $\psi$, $\parallel \psi \left(|f{|}^{2}{\parallel }_{1}\le \parallel \psi \left(20|f*{|}^{2}{\parallel }_{1}$. The special case $\psi \left(t\right)={t}^{q/2}$, $q\ge 2$, gives $\parallel f{\parallel }_{q}\le 5\parallel f*{\parallel }_{q}$ an equivalent of Littlewood.

## How to cite

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Montgomery, Hugh L.. "A note on rearrangements of Fourier coefficients." Annales de l'institut Fourier 26.2 (1976): 29-34. <http://eudml.org/doc/74283>.

@article{Montgomery1976,
abstract = {Let $f(x)\sim \Sigma a_ne^\{2\pi inx\},f*(x)\sim \sum ^\infty _\{n=0\}a*_n\,\{\rm cos\}\, 2\pi nx$, where the $a*_n$ are the numbers $\vert a_n\vert$ rearranged so that $a^*_n\searrow 0$. Then for any convex increasing $\psi$, $\Vert \psi (\vert f\vert ^2\Vert _1 \le \Vert \psi (20\vert f*\vert ^2\Vert _1$. The special case $\psi (t)=t^\{q/2\}$, $q\ge 2$, gives $\Vert f\Vert _q\le 5\Vert f*\Vert _q$ an equivalent of Littlewood.},
author = {Montgomery, Hugh L.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {29-34},
publisher = {Association des Annales de l'Institut Fourier},
title = {A note on rearrangements of Fourier coefficients},
url = {http://eudml.org/doc/74283},
volume = {26},
year = {1976},
}

TY - JOUR
AU - Montgomery, Hugh L.
TI - A note on rearrangements of Fourier coefficients
JO - Annales de l'institut Fourier
PY - 1976
PB - Association des Annales de l'Institut Fourier
VL - 26
IS - 2
SP - 29
EP - 34
AB - Let $f(x)\sim \Sigma a_ne^{2\pi inx},f*(x)\sim \sum ^\infty _{n=0}a*_n\,{\rm cos}\, 2\pi nx$, where the $a*_n$ are the numbers $\vert a_n\vert$ rearranged so that $a^*_n\searrow 0$. Then for any convex increasing $\psi$, $\Vert \psi (\vert f\vert ^2\Vert _1 \le \Vert \psi (20\vert f*\vert ^2\Vert _1$. The special case $\psi (t)=t^{q/2}$, $q\ge 2$, gives $\Vert f\Vert _q\le 5\Vert f*\Vert _q$ an equivalent of Littlewood.
LA - eng
UR - http://eudml.org/doc/74283
ER -

## References

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1. [1] G. A. BACHELIS, On the upper and lower majorant properties of Lp(G), Quart. J. Math. (Oxford), (2), 24 (1973), 119-128. Zbl0268.43003MR47 #9172
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5. [5] F. R. KEOGH, Some inequalities of Littlewood and a problem on rearrangements, J. London Math. Soc., 36 (1961), 362-376. Zbl0138.28702MR24 #A1565
6. [6] J. E. LITTLEWOOD, On a theorem of Paley, J. London Math. Soc., 29 (1954), 387-395. Zbl0058.05502MR16,126e
7. [7] J. E. LITTLEWOOD, On inequalities between f and f⋆, J. London Math. Soc., 35 (1960), 352-365. Zbl0099.05403MR24 #A799
8. [8] H. L. MONTGOMERY, Topics in multiplicative number theory, Lecture Notes in Mathematics, Springer-Verlag, Vol. 227, (1971), 187 pp. Zbl0216.03501MR49 #2616
9. [9] R. E. A. C. PALEY, Some theorems on orthogonal functions, Studia Math., 3 (1931), 226-238. Zbl0003.35201JFM57.0335.02
10. [10] H. S. SHAPIRO, Majorant problems for Fourier coefficients, to appear. Zbl0302.42004
11. [11] A. ZYGMUND, Trigonometric series, Second Edition, Cambridge University Press, 1968.

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