Coefficients of Some Double Trigonometric Cosine and Sine Series
In this paper, the results of Kano from one dimensional
cosine and sine series are extended to two dimensional cosine and sine
series. To extend these results, some classes of coefficient sequences
such as class of semi convexity and class R are extended from
one dimension to two dimensions. Further, the function f(x, y) is
two dimensional Fourier Cosine and Sine series or equivalently it
represents an integrable function or not, has been studied. Moreover,
some results are obtained which are generalization of Moricz’s
results.
[1] A. Zygmund, Trigonometric Series, Vol. I & Vol. II, Cambridge Univ.
Press, 1959.
[2] F. M´oricz, Convergence and integrability of double trigonometric series
with coefficients of bounded variation, Proc. Amer. Math. Soc., 3(1988),
633-640.
[3] F. M´oricz, Integrability of double trigonometric series with special
coefficients, Anal. Math., 16(1990), 39-56.
[4] F. M´oricz, On double cosine, sine and Walsh series with monotone
coefficients, Proc. Amer. Math. Soc., 109(2) (1990), 417-425.
[5] F. M´oricz, On the integrability and L1-convergence of double
trigonometric series, Studia Math., 98 (3) (1991), 203-225.
[6] J. Kaur and S.S. Bhatia, The Extension of the Theorem of J. W. Garrett
C. S. Rees and C. V. Stanojevic from One Dimension to Two Dimension,
International Journal of Mathematics Analysis, Vol. 3(26) (2009), 1251
- 1257.
[7] J. Kaur and S.S. Bhatia, Integrability and L1- Convergence of Double
Trigonometric Series, Analysis in Theory and Applications, vol. 27
(1)(2011), 32-39.
[8] K. Kaur, S.S. Bhatia and B. Ram, Double trigonometric series with
coefficients of bounded variation of higher order, Tamkang J. Math.,
35(4)(2004), 267-280.
[9] N.K. Bary, A treatise on trigonometric series, Vol I and Vol II, Pergamon
Press, London (1964).
[10] C.P. Chen and Y.W. Chauang, L1-convergence of double Fourier series,
Chinese J. Math., 19 (4)(1991), 391-410.
[11] S.A. Teljakovskˇii, Some estimates for trigonometric series with
quasi-convex coefficients, Mat. Sb., 63(105)(1964), 426-444.
[12] T. Kano, Coefficients of some trigonometric series, J. Fac. Sci. Shinshu
Univ., 3(1968), 153-162.
[1] A. Zygmund, Trigonometric Series, Vol. I & Vol. II, Cambridge Univ.
Press, 1959.
[2] F. M´oricz, Convergence and integrability of double trigonometric series
with coefficients of bounded variation, Proc. Amer. Math. Soc., 3(1988),
633-640.
[3] F. M´oricz, Integrability of double trigonometric series with special
coefficients, Anal. Math., 16(1990), 39-56.
[4] F. M´oricz, On double cosine, sine and Walsh series with monotone
coefficients, Proc. Amer. Math. Soc., 109(2) (1990), 417-425.
[5] F. M´oricz, On the integrability and L1-convergence of double
trigonometric series, Studia Math., 98 (3) (1991), 203-225.
[6] J. Kaur and S.S. Bhatia, The Extension of the Theorem of J. W. Garrett
C. S. Rees and C. V. Stanojevic from One Dimension to Two Dimension,
International Journal of Mathematics Analysis, Vol. 3(26) (2009), 1251
- 1257.
[7] J. Kaur and S.S. Bhatia, Integrability and L1- Convergence of Double
Trigonometric Series, Analysis in Theory and Applications, vol. 27
(1)(2011), 32-39.
[8] K. Kaur, S.S. Bhatia and B. Ram, Double trigonometric series with
coefficients of bounded variation of higher order, Tamkang J. Math.,
35(4)(2004), 267-280.
[9] N.K. Bary, A treatise on trigonometric series, Vol I and Vol II, Pergamon
Press, London (1964).
[10] C.P. Chen and Y.W. Chauang, L1-convergence of double Fourier series,
Chinese J. Math., 19 (4)(1991), 391-410.
[11] S.A. Teljakovskˇii, Some estimates for trigonometric series with
quasi-convex coefficients, Mat. Sb., 63(105)(1964), 426-444.
[12] T. Kano, Coefficients of some trigonometric series, J. Fac. Sci. Shinshu
Univ., 3(1968), 153-162.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71248", author = "Jatinderdeep Kaur", title = "Coefficients of Some Double Trigonometric Cosine and Sine Series", abstract = "In this paper, the results of Kano from one dimensional
cosine and sine series are extended to two dimensional cosine and sine
series. To extend these results, some classes of coefficient sequences
such as class of semi convexity and class R are extended from
one dimension to two dimensions. Further, the function f(x, y) is
two dimensional Fourier Cosine and Sine series or equivalently it
represents an integrable function or not, has been studied. Moreover,
some results are obtained which are generalization of Moricz’s
results.", keywords = "Conjugate Dirichlet kernel, conjugate Fejer kernel,
Fourier series, Semi-convexity.", volume = "9", number = "11", pages = "655-4", }