An inverse Sidon type inequality

S. Fridli

Studia Mathematica (1993)

  • Volume: 105, Issue: 3, page 283-308
  • ISSN: 0039-3223

Abstract

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Sidon proved the inequality named after him in 1939. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications for instance in L 1 convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality has been proved by several authors. Most of them are invariant with respect to the rearrangement of the coefficients. Although the newest results are close to best possible, no nontrivial lower estimate has been given so far. The aim of this paper is to give the best rearrangement invariant function of coefficients that can be used in a Sidon type inequality. We also show that it is equivalent to an Orlicz type and a Hardy type norm. Examples of applications are also given.

How to cite

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Fridli, S.. "An inverse Sidon type inequality." Studia Mathematica 105.3 (1993): 283-308. <http://eudml.org/doc/216000>.

@article{Fridli1993,
abstract = {Sidon proved the inequality named after him in 1939. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications for instance in $L^1$ convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality has been proved by several authors. Most of them are invariant with respect to the rearrangement of the coefficients. Although the newest results are close to best possible, no nontrivial lower estimate has been given so far. The aim of this paper is to give the best rearrangement invariant function of coefficients that can be used in a Sidon type inequality. We also show that it is equivalent to an Orlicz type and a Hardy type norm. Examples of applications are also given.},
author = {Fridli, S.},
journal = {Studia Mathematica},
keywords = {Sidon type inequalities; Hardy spaces; convergence classes; Hardy space; trigonometric Dirichlet kernels; best rearrangement invariant function of coefficients; Sidon type inequality},
language = {eng},
number = {3},
pages = {283-308},
title = {An inverse Sidon type inequality},
url = {http://eudml.org/doc/216000},
volume = {105},
year = {1993},
}

TY - JOUR
AU - Fridli, S.
TI - An inverse Sidon type inequality
JO - Studia Mathematica
PY - 1993
VL - 105
IS - 3
SP - 283
EP - 308
AB - Sidon proved the inequality named after him in 1939. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications for instance in $L^1$ convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality has been proved by several authors. Most of them are invariant with respect to the rearrangement of the coefficients. Although the newest results are close to best possible, no nontrivial lower estimate has been given so far. The aim of this paper is to give the best rearrangement invariant function of coefficients that can be used in a Sidon type inequality. We also show that it is equivalent to an Orlicz type and a Hardy type norm. Examples of applications are also given.
LA - eng
KW - Sidon type inequalities; Hardy spaces; convergence classes; Hardy space; trigonometric Dirichlet kernels; best rearrangement invariant function of coefficients; Sidon type inequality
UR - http://eudml.org/doc/216000
ER -

References

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  1. [1] R. Bojanic and Č. Stanojević, A class of L 1 -convergence, Trans. Amer. Math. Soc. 269 (1982), 677-683. Zbl0493.42011
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  8. [8] F. Móricz, Sidon type inequalities, Publ. Inst. Math. (Beograd) 48 (62) (1990), 101-109. Zbl0727.42005
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  12. [12] S. Sidon, Hinreichende Bedingungen für den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc. 14 (1939), 158-160. Zbl65.0255.02
  13. [13] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. Zbl0232.42007
  14. [14] N. Tanović-Miller, On integrability and L 1 convergence of cosine series, Boll. Un. Mat. Ital. (7) 4-B (1990), 499-516. Zbl0725.42007
  15. [15] S. A. Telyakovskiǐ, Concerning a sufficient condition of Sidon for the integrability of trigonometric series, Mat. Zametki 14 (1973), 317-328 (in Russian); English transl.: Math. Notes 14 (1973), 742-748. 
  16. [16] S. A. Telyakovskiǐ, On the integrability of sine series, Trudy Mat. Inst. Steklov. 163 (1984), 229-233 (in Russian); English transl.: Proc. Steklov Inst. Mat. 4 (1985), 269-273. 
  17. [17] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. (2) 12 (1913), 41-70. Zbl44.0300.03

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