Open Access

On the mean summability by Cesaro method of Fourier trigonometric series in two-weighted setting

Journal of Inequalities and Applications20062006:41837

DOI: 10.1155/JIA/2006/41837

Received: 26 June 2005

Accepted: 23 October 2005

Published: 27 April 2006


The Cesaro summability of trigonometric Fourier series is investigated in the weighted Lebesgue spaces in a two-weight case, for one and two dimensions. These results are applied to the prove of two-weighted Bernstein's inequalities for trigonometric polynomials of one and two variables.


Authors’ Affiliations

Department of Mathematics, Faculty of Art and Science, Balikesir University
International Black Sea University


  1. Banach S, Steinhaus H: Sur le principe de condensation de singularités. Fundamenta Mathematicae 1927, 9: 50–61.MATHGoogle Scholar
  2. Kokilashvili V, Krbec M: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, New Jersey; 1991:xii+233.MATHView ArticleGoogle Scholar
  3. Muckenhoupt B: Weighted norm inequalities for the Hardy maximal function. Transactions of the American Mathematical Society 1972, 165: 207–226.MATHMathSciNetView ArticleGoogle Scholar
  4. Muckenhoupt B: Two weight function norm inequalities for the Poisson integral. Transactions of the American Mathematical Society 1975, 210: 225–231.MATHMathSciNetView ArticleGoogle Scholar
  5. Nakhman AD, Osilenker BP: Estimates of weighted norms of some operators generated by multiple trigonometric Fourier series. Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika 1982,239(4):39–50.MathSciNetGoogle Scholar
  6. Rosenblum M: Summability of Fourier series in. Transactions of the American Mathematical Society 1962,105(1):32–42.MATHMathSciNetGoogle Scholar
  7. Tsanava Ts: On the Fourier operators in weighted Lebesgue spaces. Proceedings of A. Razmadze Mathematical Institute 2005, 138: 107–109.Google Scholar
  8. Žižiašvili LV: Conjugate Functions and Trigonometric Series. Izdat. Tbilis. Univ., Tbilisi; 1969:271.Google Scholar
  9. Zygmund A: Trigonometric Series. Vols. I, II. 2nd edition. Cambridge University Press, New York; 1959:Vol. I. xii+383; Vol. II. vii+354.Google Scholar


© A. Guven and V. Kokilashvili. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.