Skip to main content

On Approximation of Signals in the Weighted Zygmund Class via \((E,1)(\overline{N},p_{n})\) Summability Means of Conjugate Fourier Series

  • Conference paper
  • First Online:
Recent Advances in Intelligent Information Systems and Applied Mathematics (ICITAM 2019)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 863))

  • 591 Accesses

Abstract

Approximation of signals in Lipschitz and Zygmund classes via different summability means have been demonstrated by many researchers. In the proposed paper, we have studied an estimation of the order of convergence of conjugate Fourier series in the weighted Zygmund class \(W(Z_{r}^{(\omega )})\) by using \((E,1)(\overline{N},p_{n})\)-product summability means and accordingly established some new results. Also, the results obtained here generalizes some known results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aasma, A., Dutta, H., Natarajan, P.N.: An Introductory Course in Summability Theory, 1st edn. Wiley, Hoboken (2017)

    Book  Google Scholar 

  2. Das, A.A., Jena, B.B., Paikray, S.K., Jati, R.K.: Statistical deferred weighted summability and associated Korovokin-type approximation theorem. Nonlinear Sci. Lett. A 9(3), 238–245 (2018)

    Google Scholar 

  3. Dutta, H., Rhoades, B.E. (eds.): Current Topics in Summability Theory and Applications, 1st edn. Springer, Singapore (2016)

    MATH  Google Scholar 

  4. Hardy, G.H.: Divergent Series, 1st edn. Oxford University Press, Oxford (1949)

    MATH  Google Scholar 

  5. Jena, B.B., Paikray, S.K., Misra, U.K.: Inclusion theorems on general convergence and statistical convergence of \((L,1,1)\) - summability using generalized Tauberian conditions. Tamsui Oxford J. Inf. Math. Sci. 31, 101–115 (2017)

    MathSciNet  Google Scholar 

  6. Jena, B.B., Vandana, Paikray, S.K., Misra, U.K.: On generalized local property of \(| A;\delta | _ {k} \)-summability of factored Fourier series. Int. J. Anal. Appl. 16, 209–221 (2018)

    Article  Google Scholar 

  7. Jena, B.B., Paikray, S.K., Misra, U.K.: Statistical deferred Cesàro summability and its applications toapproximation theorems. Filomat 32, 2307–2319 (2018)

    Article  MathSciNet  Google Scholar 

  8. Jena, B.B., Mishra, L.N., Paikray, S.K., Misra, U.K.: Approximation of signals by general matrix summability with effects of Gibbs phenomenon. Bol. Soc. Paran. Mat. (2018). https://doi.org/10.5269/bspm.v38i6.39280

    Article  MathSciNet  Google Scholar 

  9. Jena, B.B., Paikray, S.K., Misra, U.K.: A Tauberian theorem for double Cesàro summability method. Int. J. Math. Math. Sci. 2016, 1–4 (2016). Article ID 2431010

    Article  Google Scholar 

  10. Lal, S., Shireen: Best approximation of functions of generalized Zygmund class by matrix-Euler summability mean of fourier series. Bull. Math. Anal. Appl. 5, 1–13 (2013)

    Google Scholar 

  11. Lal, S., Mishra, A.: Euler-Hausdörff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to the generalized Lipschitz class. J. Inequal. Appl. 2013, 1–14 (2013). Article ID 59

    Google Scholar 

  12. Leindler, L.: Strong approximation and generalized Zygmund class. Acta Sci. Math. 43, 301–309 (1981)

    MathSciNet  MATH  Google Scholar 

  13. Misra, U.K., Sahoo, N.C., Paikray, S.K.: Summability of a series by \(Y-|\bar{N}, q_{n}|_{k}\)-method. Indian J. Math. Math. Sci. 3, 117–126 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Misra, U.K., Sahoo, N.C., Paikray, S.K.: A Note on \(|N, p_{n}^{\alpha },\delta |_{k}\)-summability. Indian Acad. Math. 30, 481–487 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Móricz, F.: Enlarged Lipschitz and Zygmund classes of functions and fourier transforms. East J. Approx. 16, 259–271 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Móricz, F., Németh, J.: Generalized Zygmund classes of functions and strong approximation by fourier series. Acta Sci. Math. 73, 637–647 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Parida, P., Paikray, S.K., Dutta, H., Jena, B.B., Dash, M.: Tauberian theorems for Cesàro summability of \(n\)-thsequences. Filomat 32, 3993–4004 (2018)

    Article  MathSciNet  Google Scholar 

  18. Parida, P., Paikray, S.K., Dash, M., Misra, U.K.: Degree of approximation by product \((\overline{N}, p_{n}, q_{n})(E, q)\)-summability of Fourier series of a signal belonging to \(Lip(\alpha, r)\)-class. TWMS J. App. Eng. Math. 9, 901–908 (2019)

    Google Scholar 

  19. Paikray, S.K., Misra, U.K., Sahoo, N.C.: Trangular matrix summability of a series. African J. Math. Comput. Sci. Res. 4, 164–169 (2011)

    Google Scholar 

  20. Pradhan, T., Paikray, S.K., Jena, B.B., Dutta, H.: Statistical deferred weighted \(\cal{B} \)-summability and its applications to associated approximation theorems. J. Inequal. Appl. 2018, 1–21 (2018). Article ID: 65

    Article  MathSciNet  Google Scholar 

  21. Singh, M.V., Mittal, M.L., Rhoades, B.E.: Approximation of functions in the generalized Zygmundclass using Hausdorff means. J. Inequal. Appl. 2017 (2017). https://doi.org/10.1186/s13600-017-1361-8

  22. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: A certain class of weighted statistical convergence and associatedKorovkin-type approximation theorems for trigonometric functions. Math. Methods Appl. Sci. 41, 671–683 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. (RACSAM) 112, 1487–1501 (2017)

    Article  Google Scholar 

  24. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: Deferred weighted \(\cal{A}\)-statistical convergence based upon the \((p, q)\)-Lagrange polynomials and its applications to approximation theorems. J. Appl. Anal. 21, 1–16 (2018)

    Article  MathSciNet  Google Scholar 

  25. Titechmalch, E.C.: The Theory of Functions. Oxford University Press, Oxford (1939)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. K. Paikray .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Das, A.A., Paikray, S.K., Jati, R.K. (2020). On Approximation of Signals in the Weighted Zygmund Class via \((E,1)(\overline{N},p_{n})\) Summability Means of Conjugate Fourier Series. In: Castillo, O., Jana, D., Giri, D., Ahmed, A. (eds) Recent Advances in Intelligent Information Systems and Applied Mathematics. ICITAM 2019. Studies in Computational Intelligence, vol 863. Springer, Cham. https://doi.org/10.1007/978-3-030-34152-7_61

Download citation

Publish with us

Policies and ethics