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Summation methods applied to Voronovskaya-type theorems for the partial sums of Fourier series and for Fejér operators

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Published/Copyright: April 26, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 1

Abstract

We obtain Voronovskaya-type theorems for the partial sums of Fourier series using the second order Cesáro method of summation. Then we obtain two versions of Voronovskaya-type theorems for Fejér operators and finally we deduce an integral identity.

Keywords: partial sums of Fourier series; Fejér operators; Voronovskaya theorem; Cesáro summations; the method of summation of Bosanquet and Linfoot; conjugate Fourier series
MSC: Primary 42A20; 42A24; Secondary 42A50; 40G05; 40G99; 41A36

The paper was partially supported by the Sectoral Operational Programme Human Resources Development (SOP HRD), financed from the European Social Fund and by the Romanian Government under the contract number POSDRU/88/1.5/S/59321

Acknowledgement

The authors are grateful to the anonymous referee for the careful reading of the manuscript.

  1. This paper has been Communicated by Stanisława Kanas

References

[1] Bosanquet, L. S.—Linfoot, E. H.: Generalized means and the summability of Fourier series, Q. J. Math. 2 (1931), 207–229.10.1093/qmath/os-2.1.207Search in Google Scholar

[2] Petersen, G. M.: Regular Matrix Transformations, McGraw-Hill, London, 1966.Search in Google Scholar

[3] Smith, A. H.: On the summability of derived conjugate series of the Fourier-Lebesgue type, Bull. Amer. Math. Soc. 40 (1934), 406–412.10.1090/S0002-9904-1934-05873-7Search in Google Scholar

[4] Voronovskaya, E. V.: Determination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein, C. R. Acad. Sci. URSS (1932), 79–85 (Russian).Search in Google Scholar

[5] Zygmund, A.: Trigonometric Series} (2nd ed.), University Press, Cambridge, 1959.Search in Google Scholar

Received: 2012-8-2
Accepted: 2013-7-20
Published Online: 2016-4-26
Published in Print: 2016-2-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0131
Keywords for this article
partial sums of Fourier series; Fejér operators; Voronovskaya theorem; Cesáro summations; the method of summation of Bosanquet and Linfoot; conjugate Fourier series

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  2. Prof. RNDr. Ján Jakubík, DrSc. passed away
  3. Research Article
  4. On the number of slim, semimodular lattices
  5. Research Article
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  7. Research Article
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  9. Research Article
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  13. Research Article
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