DOI QR코드

DOI QR Code

On Absolute Almost Generalized Nörlund Summability of Orthogonal Series

  • Received : 2011.02.04
  • Accepted : 2011.11.30
  • Published : 2012.09.23

Abstract

In this paper we present some results on absolute almost generalized N$\ddot{o}$rlund summability of orthogonal series. The most important corollaries of the main results also are deduced.

Keywords

References

  1. G. Alexits, Convergence problems of orthogonal series, Translated from the German by I. Folder. International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York-Oxford-Paris 1961, 350 pp.
  2. P. Billard, Sur la sommabilite absolue des series de functions orthogonales, (French) Bull. Sci. Math., 85(1961), 29-33.
  3. T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc., 7(1957), 113-141. https://doi.org/10.1112/plms/s3-7.1.113
  4. L. V. Grepachevskaya, Absolute summability of orthogonal series, (Russian) Mat. Sb. (N. S.), 65(107)(1964), 370-389.
  5. G. H. Hardy, Divergent Series, first edition, Oxford University Press, 1949.
  6. Xh. Z. Krasniqi, A note on |N,p,q|k $(1\;{\leq}\;K\;{\leq}\;2)$ summability of orthogonal series, Note Mat., 30(2)(2010), 135-139.
  7. Xh. Z. Krasniqi, On absolute generalized Norlund summability of double orthogonal series, Int. J. Nonlinear Anal. Appl., 2(2)(2011), 63-75.
  8. Xh. Z. Krasniqi, On absolutely almost convergency of higher order of orthogonal series, Int. J. Open Problems Comput. Sci. Math., 4(1)(2011), 44-51.
  9. Xh. Z. Krasniqi, Certain sufficient conditions on $|N,p_{n},q_{n}|k$ summability of orthogonal series, J. Nonlinear Sci. Appl., (to appear).
  10. Xh. Z. Krasniqi, On absolute weighted mean summability of orthogonal series, Selcuk J. Appl. Math. Appl., 12(2)(2011), 63-70.
  11. Xh. Z. Krasniqi, On $|A,{\delta}|_k$-summability of orthogonal series, Math. Bohem., 137(1)(2012), 17-25.
  12. L. Leindler, Uber die absolute summierbarkeit der orthogonalreihen, (German) Acta Sci. Math. (Szeged), 22(1961), 243-268.
  13. L. Leindler, On the absolute Riesz summability of orthogonal series, Acta Sci. Math. (Szeged), 46(1-4)(1983), 203-209.
  14. L. Leindler and K. Tandori, On absolute summability of orthogonal series, Acta Sci. Math. (Szeged), 50(1-2)(1986), 99-104.
  15. L. Leindler, On the newly generalized absolute Riesz summability of orthogonal series, Anal. Math., 21(4)(1995), 285-297. https://doi.org/10.1007/BF01909151
  16. G. G. Lorentz, A contribution to the theory of divergent series, Acta Math., 80(1948), 167-190. https://doi.org/10.1007/BF02393648
  17. S. M. Mazhar, A. H. Siddiqi, On olmost summability of a trigonometric sequence, Acta Math. Acad. Sci. Hungar., 20(1969), 21-24. https://doi.org/10.1007/BF01894565
  18. Y. Okuyama, On the absolute Norlund summability of orthogonal series, Proc. Japan Acad., 54(1978), 113-118. https://doi.org/10.3792/pjaa.54.113
  19. Y. Okuyama, On the absolute Norlund summability of orthogonal series, Proc. Japan Acad. Ser. A Math. Sci., 54(5)(1978), 113-118. https://doi.org/10.3792/pjaa.54.113
  20. Y. Okuyama and T. Tsuchikura, On the absolute Riesz summability of orthogonal series, Analysis Math., 7(1981), 199-208. https://doi.org/10.1007/BF01908522
  21. Y. Okuyama, Absolute summability of Fourier series and orthogonal series. Lecture Notes in Mathematics, 1067. Springer-Verlag, Berlin, 1984, 118 pp.
  22. Y. Okuyama, On the absolute Riesz summability of orthogonal series, Tamkang J. Math. 19(3)(1988), 75-89.
  23. Y. Okuyama, On the absolute generalized Norlund summability of orthogonal series, Tamkang J. Math., 33(2002), 161-165.
  24. K. Qureshi, On the degree of approximation of a periodic function f by almost Norlund means, Tamkang J. Math., 12(1)(1981), 35-38.
  25. V. N. Spevakov, B. A. Kudrjavcev, Absolute summability of orthogonal series by the Euler method, (Russian) Mat. Zametki, 21(1)(1977), 51-56.
  26. I. Szalay, On generalized absolute Cesaro summability of orthogonal series, Acta Sci. Math. (Szeged), 32(1971), 51-57.
  27. K. Tandori, Uber die orthogonalen Funktionen IX (Absolute Summation), Acta Sci. Math. (Szeged), 21(1960), 292-299.
  28. M. Tanaka, On generalized Norlund methods of summability, Bull. Austral. Math. Soc., 19(1978), 381-402. https://doi.org/10.1017/S0004972700008935
  29. P. L. Ulyanov, Solved and unsolved problem in the theory of trigonometric and orthogonal series, Uspehi Math. Nauk., 19(1964), 3-69 .