A note on rearrangements of Fourier coefficients

Hugh L. Montgomery

Annales de l'institut Fourier (1976)

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

Abstract

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Let f ( x ) ∼ Σ a n e 2 π i n x , f * ( x ) ∼ ∑ n = 0 ∞ 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 ≤ ∥ 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.

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
  2. [2] A. BAERNSTEIN, II, Integral means, univalent functions and circular symmetrizations, Acta Math., 133 (1974), 139-169. Zbl0315.30021MR54 #5456
  3. [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. Zbl0001.13504JFM57.0315.01
  4. [4] G. H. HARDY and J. E. LITTELWOOD, A new proof of a theorem on rearrangements, J. London math. Soc., 23 (1949), 163-168. Zbl0034.04301MR10,448b
  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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