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  • Research Article
  • Open Access

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

Journal of Inequalities and Applications20062006:41837

  • Received: 26 June 2005
  • Accepted: 23 October 2005
  • Published:


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.


  • Fourier Series
  • Trigonometric Polynomial
  • Lebesgue Space
  • Trigonometric Series
  • Weighted Lebesgue Space


Authors’ Affiliations

Department of Mathematics, Faculty of Art and Science, Balikesir University, Balikesir, 10145, Turkey
International Black Sea University, Tbilisi, 0131, Georgia


  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