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Trigonometric Fourier Approximation of the Conjugate Series of a Function of Generalized Lipchitz Class by Product Summability

  • B. P. Padhy
  • P. K. Das
  • M. Misra
  • P. Samanta
  • U. K. Misra
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 410)

Abstract

Trigonometric Fourier approximation and Lipchitz class of function had been introduced by Zygmund and McFadden respectively. Dealing with degree of approximation of conjugate series of a Fourier series of a function of Lipchitz class Misra et al. have established certain theorems. Extending their results, in this paper a theorem on trigonometric approximation of conjugate series of Fourier series of a function \( f \in \,Lip\,\left( {\alpha ,r} \right) \) by product summability (E, s) (N, p n , q n ) has been established.

Keywords

Fourier approximation \( Lip\,\left( {\alpha ,r} \right) \) class of function (E, q) mean (N, pn, qn) mean (E, s) (N, pn, qn) product mean Conjugate fourier series Lebesgue integral 

2010-Mathematics Subject Classification:

42B05 42B08 

References

  1. 1.
    Borwein, D.: On product of sequences. J. London Math. Soc. 33, 352–357 (1958)Google Scholar
  2. 2.
    Hardy, G.H.: Divergent Series (First Edition). Oxford University Press, Oxford (1970)Google Scholar
  3. 3.
    Mcfadden, L.: Absolute Norlund summabilty. Duke Math. J. 9, 168–207 (1942)Google Scholar
  4. 4.
    Misra, U.K., Misra, M., Padhy, B.P., Buxi, S.K.: On degree of approximation by product means of conjugate series of Fourier series. Int. J. Math. Sci. Eng. Appl. 6(1), 363–370 (2012). ISSN 0973–9424Google Scholar
  5. 5.
    Misra, U.K., Paikray, S.K., Jati, R.K, Sahoo, N.C.: On degree of Approximation by product means of conjugate series of Fourier series. Bull. Soc. Math. Serv. Stan. 1(4), 12–20 (2012). ISSN 2277–8020Google Scholar
  6. 6.
    Titchmarch, E.C.: The Theory of Functions. Oxford University Press, Oxford (1939)Google Scholar
  7. 7.
    Zygmund, A.: Trigonometric Series (Second Edition), vol. I. Cambridge University Press, Cambridge (1959)Google Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

  • B. P. Padhy
    • 1
  • P. K. Das
    • 2
  • M. Misra
    • 3
  • P. Samanta
    • 4
  • U. K. Misra
    • 5
  1. 1.Department of Mathematics, School of Applied SciencesKIIT UniversityBhubaneswarIndia
  2. 2.Biswasray Science CollegeGanjamIndia
  3. 3.Department of MathematicsB.A. CollegeBerhampurIndia
  4. 4.Department of MathematicsBerhampur UniversityBerhampurIndia
  5. 5.National Institute of Scince and TechnologyGanjamIndia

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