Download PDF - An inverse Sidon type inequalityAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

An inverse Sidon type inequality

Authors 1

Affiliations

  1. Department of Numerical Analysis, L. Eötvös University, Bogdánfy U. 10/B, H-1117 Budapest, Hungary.

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 L1 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.

Keywords

ENG
Sidon type inequalities, Hardy spaces, convergence classes

Bibliography

  1. R. Bojanic and Č. Stanojević, A class of L1-convergence, Trans. Amer. Math. Soc. 269 (1982), 677-683.
  2. M. Buntinas and N. Tanović-Miller, New integrability and L1-convergence classes for even trigonometric series, Rad. Mat. 6 (1990), 149-170.
  3. M. Buntinas and N. Tanović-Miller, New integrability and L1-convergence classes for even trigonometric series II, in: Approximation Theory, Kecskemét 1990, Colloq. Math. Soc. János Bolyai 58, North-Holland, 1991, 103-125.
  4. G. A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), 213-222 (in Russian); English transl.: Math. Notes 23 (1978), 117-123.
  5. B. S. Kashin and A. A. Saakyan, Orthogonal Series, Nauka, Moscow 1984.
  6. A. N. Kolmogorov, Sur l'ordre de grandeur des coefficients de la série de Fourier-Lebesgue, Bull. Internat. Acad. Polon. Sci. Lettres Sér. (A) Sci. Math. 1923, 83-86.
  7. M. A. Krasnosel'skiǐ and Ya. B. Rutickiǐ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen 1961.
  8. F. Móricz, Sidon type inequalities, Publ. Inst. Math. (Beograd) 48 (62) (1990), 101-109.
  9. F. Móricz, On the integrability and L1-convergence of sine series, Studia Math. 92 (1989), 187-200.
  10. F. Schipp, Sidon-type inequalities, in: Approximation Theory, Lecture Notes in Pure and Appl. Math. 138, Dekker, New York 1991, 421-436.
  11. F. Schipp, W. R. Wade and P. Simon (with assistance from J. Pál), Walsh Series, Hilger, Bristol 1990.
  12. S. Sidon, Hinreichende Bedingungen für den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc. 14 (1939), 158-160.
  13. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
  14. N. Tanović-Miller, On integrability and L1 convergence of cosine series, Boll. Un. Mat. Ital. (7) 4-B (1990), 499-516.
  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. 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. W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. (2) 12 (1913), 41-70.
Studia Mathematica
1993/105/3
Export to PBN, Opens in new tab
Pages:
283-308
Main language of publication
English
Received
1992-12-14
Published
1993
Exact and natural sciences