Advertisement

On the degree of approximation of functions in a weighted Lipschitz class by almost matrix summability method

Proceedings: ICMAA 2016
  • 176 Downloads

Abstract

In this paper, we obtain the degree of approximation of functions belonging to the weighted Lipschitz class \(W(L^p,\xi (t))\) and their conjugates through almost matrix means of their Fourier series and conjugate Fourier series, respectively. We also derive some corollaries from our theorems.

Keywords

Fourier series Conjugate function Degree of approximation Weighted norm Generalized Minkowski inequality and Almost matrix means 

Mathematics Subject Classification

41A25 26A15 40A35 

Notes

Acknowledgements

Authors are thankful to the reviewers for their valuable suggestions for improvement of the manuscript.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no competing interests.

References

  1. 1.
    Zygmund, A. 2002. Trigonometric series. Cambridge: Cambridge University Press.MATHGoogle Scholar
  2. 2.
    Khan, H.H. 1982. A note on a theorem of Izumi. Comm. Fac. Sci. Math. Ankara(Turkey). 31:123–127.Google Scholar
  3. 3.
    Lorentz, G.G. 1948. A contribution to the theory of divergent series. Acta Math. 80: 167–190.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Mazhar, S.M., and A.H. Siddiqui. 1969. On almost summability of a trigonometric sequence. Acta Math. Acad. Sci. Hungar. 20: 21–24.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Silverman, L.L. 1913. On the definition of the sum of a divergent series, The University of Missouri Studies, Mathematics Series, 1.Google Scholar
  6. 6.
    Toeplitz, O. 1911. \(\ddot{U}\)ber allgemeine linear mittebildungen. Prace Mathematyczno Fizyczne 22: 113–119.Google Scholar
  7. 7.
    Singh, U., and S.K. Srivastava. 2014. Trigonometric approximation of functions belonging to certain Lipschitz classes by \(C^1T\) operator. Asian-European J. Maths. 7 (4): 1–13.Google Scholar
  8. 8.
    King, J.P. 1966. Almost summable sequences. Proc. Amer. Math. Soc. 17: 1219–1225.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nanda, S. 1976. Some sequence spaces and almost convergence, J. Aust. Math. Soc Series A 22:446–455.Google Scholar
  10. 10.
    Sharma, P.L., and K. Qureshi. 1980. On the degree of approximation of a periodic function by almost Riesz means. Ranchi Univ. Math. 11: 29–43.MathSciNetMATHGoogle Scholar
  11. 11.
    Qureshi, K. 1981. On the degree of approximation of a periodic function \(f\) by almost N\(\ddot{o}rlund\) means. Tamkang J. Math. 12 (1): 35–38.MathSciNetGoogle Scholar
  12. 12.
    Lal, S. 2004. On the approximation of function belnging to weighted \(W(L^p,\xi (t))\) class by almost matrix summability method of its Fourier series. Tamkang J. Math. 35 (1): 67–76.MathSciNetMATHGoogle Scholar
  13. 13.
    Khan, H.H. 1973–74. On the degree of approximation of function belonging to weighted \((L^p, \psi (t))\), The Alig. Bull. of Maths, 3–4:83–88.Google Scholar
  14. 14.
    Mittal, M.L., Mishra, V.N. 2008. Approximation of signals(functions) belnging to weighted \(W(L^p,\xi (t)),(p\ge 1)-\)class by almost matrix summability method of its Fourier series, International J. of Math. Sci. and Engg. Appls.(IJMSEA), 2, (4): 285–294.Google Scholar
  15. 15.
    Mishra, V.N. 2009. On the degree of approximation of signals(functions) belnging to weighted \(W(L^p,\xi (t)),(p\ge 1)-\)class by almost matrix summability method of its conjugate Fourier series. Int. J. of Appl. Math and Mech. 5 (7): 16–27.Google Scholar
  16. 16.
    Mishra, V.N., H.H. Khan, I.A. Khan, and L.N. Mishra. 2014. On the degree of approximation of signals \(Lip(\alpha, p), (p\ge 1)\) class by almost Riesz means of its Fourier series. J. Class. Anal. 4 (1): 79–87.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Deepmala, Piscoran. 2016. LI, Approximation of signals(functions) belonging to certain Lipschitz classes by almost Riesz means of its Fourier series. J. Inequal. Appl. 2016: 163. doi: 10.1186/s13660-016-1101-5.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

Personalised recommendations