Convergence of Double Cosine Series
Download PDF
Authors: K. SINGH AND K. MODI
DOI: 10.46793/KgJMat2003.443S
Abstract:
In this paper we consider double cosine series whose coefficients form a null sequence of bounded variation of order (p, 0), (0,p) and (p,p) with the weight (jk)p−1 for some p > 1. We study pointwise convergence, uniform convergence and convergence in Lr-norm of the series under consideration. In a certain sense our results extend the results of Young , Kolmogorov and Móricz.
Keywords:
Rectangular partial sums, Lr−convergence, Cesàro means, monotone sequences.
References:
[1] C. P. Chen and Y. W. Chauang, L1-convergence of double Fourier series, Chin. J. Math. 19(4) (1991), 391–410.
[2] K. Kaur, S. S. Bhatia and B. Ram, L1-Convergence of complex double trigonometric series, Proc. Indian Acad. Sci. 113(3) (2003), 1–9.
[3] A. N. Kolmogorov, Sur l’ordre de grandeur des coefficients de la sėrie de Fourier-Lebesque, Bulletin L’Academie Polonaise des Science (1923), 83–86.
[4] F. Móricz, Convergence and integrability of double trigonometric series with coefficients of bounded variation, Proc. Amer. Math. Soc. 102 (1988), 633–640.
[5] F. Móricz, On the integrability and L1-convergence of double trigonometric series, Studia Math. (1991), 203–225.
[6] T. M. Vukolova, Certain properties of trigonometric series with monotone coefficients (English, Russian original), Moscow Univ. Math. Bull. 39(6) (1984), 24–30, translation from Vestnik Moskov. Univ. 1(6) (1984), 18–23.
[7] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. 12(2) (1913), 41–70.
[8] A. Zygmund, Trigonometric Series, Vols. I, II, Cambridge University Press, Cambridge, 1959.
Login:
