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Generalized Absolute Riesz Summability of Orthogonal Series

  • K. KalaivaniEmail author
  • C. Monica
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In this paper, for 1 ≤ k ≤ 2 and a sequence \(\gamma :=\{\gamma (n)\}_{n=1}^{\infty }\) that is quasi β-power monotone decreasing with \(\beta >1-\dfrac {1}{k},\) we prove the |A, γ|k summability of an orthogonal series, where A is Riesz matrix. For \(\beta >\frac {1}{2},\) we give a necessary and sufficient condition for |A, γ|k summability, where A is Riesz matrix. Our result generalizes the result of Moricz (Acta Sci Math 23:92–95, 1962) for absolute Riesz summability of an orthogonal series.

References

  1. 1.
    Ferenc Moricz.: \(\ddot {U}\)ber die Rieszsche Summation der Orthogonal reihen. Acta Sci. Math. 23, 92–95 (1962).Google Scholar
  2. 2.
    Leindler, L.: On the converse of inequalities of Hardy and Littlewood. Acta Sci. Math. 58, 191–196 (1993).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Leindler, L., Németh, J.: On the connection of quasi power-monotone and quasi geomentrical sequences with application to integrability theorems for power series. Acta Math. Hungar. 68, 7–19 (1995).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Leindler, L.: On the newly generalized absolute Cesaro summability of orthogonal series. Acta Math. Hungar. 68, 295–316 (1995).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zygmund, A.: Trigonometric Series. I, Cambridge. (1959).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Vellore Institute of TechnologyVelloreIndia

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