On Generalized Local Property of $|A;\delta|_{k}$-Summability of Factored Fourier Series

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B. B. Jena
Vandana --
S. K. Paikray
U. K. Misra

Abstract

The convergence of Fourier series of a function at a point depends upon the behaviour of the function in the neighborhood of that point and it leads to the local property of Fourier series. In the proposed paper a new result on local property of $|\mathcal{A};\delta|_{k}$-summability of factored Fourier series has been established that generalizes a theorem of Sarig\"{o}l [13] (see [M. A. Sari\"{o}gol, On local property of $|\mathcal{A}|_{k}$-summability of factored Fourier series, \textit{J. Math. Anal. Appl.} 188 (1994), 118-127]) on local property of $|\mathcal{A}|_{k}$-summability of factored Fourier series.

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References

  1. S. N. Bhatt, An aspect of local property of |R,log,1| summability of the factored Fourier series, Proc. Natl. Inst. India 26 (1960), 69-73.
  2. H. Bor, A note on local property of factored Fourier series, J. Non. Anal. 64 (2006), 513-517.
  3. H. Bor, Local property of | ¯ N,p n | k -summability of factored Fourier series, Bull. Inst. Math. Acad. Sinica 17 (1989), 165-170.
  4. H. Bor, On the local property of | ¯ N,p n | k -summability of factored Fourier series, J. Math. Anal. 163 (1992), 220-226.
  5. Deepmala, Piscoran Laurian-Ioan, Approximation of signals (functions) belonging to certain Lipschitz classes by almost Riesz means of its Fourier series, J. Inequal. Appl. 2016 (2016), Art. ID 163.
  6. T. M. Fleet, On an extension of absolute summability and some theorems of Littlewood and paley, Proc. London Math. Soc. 37 (1957), 113-141.
  7. K. Matsumoto, Local property of the summability |R,p n ,1|, Tohoku Math. J 8 (1956), 114-124.
  8. K. N. Mishra, Multipliers for | ¯ N,p n |-summability of Fourier series, Bull.Inst. Math. Acad. Sinica 14 (1984), 431-438.
  9. V. N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India.
  10. V. N. Mishra, L. N. Mishra, Trigonometric Approximation of Signals (Functions) in L p (p ≥ 1)- norm, Int. J. Contem. Math. Sci. 7 (2012), 909-918.
  11. V. N. Mishra, S. K. Paikray, P. Palo, P. N. Samanta, M. Misra, U. K. Misra, On double absolute factorable matrix summability, Tbilisi Math. J. 10 (2017), 29-44.
  12. R. Mohanty, On the summability |R,logw,1| of Fourier series, J. London Math. soc. 25 (1950), 67-72.
  13. M. A. Sariögol, On local property of |A| k -summability of factored Fourier series, J. Math. Anal. Appl. 188 (1994), 118-127.
  14. W. T. Sulaiman, On local property of absolute weighted mean summability of Fourier series, Bull. Math. Anal. Appl. 4 (2011), 163-168.
  15. D. S Yu and P. Zhou, A new class of matrices and its applications to absolute summability factor theorems, Math. Comput. Model. 57 (2013), 401-412.
  16. E. C. Tichmarsh, The Theory of Functions, Oxford University Press, London, (1961).
  17. A. Zygmund, Trigonometric series, vol. I, Cambridge Univ. Press, Cambridge, (1959).