On Some Sufficient Conditions for L1-Convergence of Double Sine Series

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Abstract

In this paper we introduce some numerical classes of double sequences. Such classes are used to show some sufficient conditions for L1 −convergence of double sine series. This study partially extends very recent results of Leindler, and particularly those of Zhou, from single to two-dimensional sine series.

Abstract

In this paper we introduce some numerical classes of double sequences. Such classes are used to show some sufficient conditions for L1 −convergence of double sine series. This study partially extends very recent results of Leindler, and particularly those of Zhou, from single to two-dimensional sine series.

References

  • 1. Bokayev, N.A.; Mukanov, Zh.B. - Weighted integrability of double trigonometric series and of double series with respect to multiplicative systems with coefficients of class R+ 0 BV S2, Translation of Mat. Zametki, 91 (2012), 617-620. Math. Notes, 91 (2012), 575-578.

  • 2. Chen, C.P. - Integrability and L-convergence of multiple trigonometric series, Bull. Austral. Math. Soc., 49 (1994), 333-339.

  • 3. Chen, C.P.; Wu, H.C.; M´oricz, F. - Pointwise convergence of multiple trigono- metric series, J. Math. Anal. Appl., 185 (1994), 629-646.

  • 4. Chen, C.P.; Chauang, Y.W. - L1-convergence of double Fourier series, Chinese J. Math., 19 (1991), 391-410.

  • 5. Kaur, K.; Bhatia, S.S.; Ram, B. - L1-convergence of complex double Fourier series, Proc. Indian Acad. Sci. Math. Sci., 113 (2003), 355-363.

  • 6. Kaur, J.; Bhatia, S.S. - Integrability and L1-convergence of double cosine trigono- metric series, Anal. Theory Appl., 27 (2011), 32-39.

  • 7. Leindler, L. - On the degree of approximation of continuous functions, Acta Math. Hungar., 104 (2004), 105-113.

  • 8. Leindler, L. - Embedding results regarding strong approximation of sine series, Acta. Sci. Math. (Szeged), 71 (2005), 91-103.

  • 9. Leindler, L. - On L1 −convergence of sine series, Anal. Math., 38 (2012), 123-133.

  • 10. Móricz, F. - On the integrability and L1- convergence of double trigonometric series, Studia Math., 98 (1991), 203-225.

  • 11. Móricz, F. - Convergence and integrability of double trigonometric series with coeffi- cients of bounded variation, Proc. Amer. Math. Soc., 102 (1988), 633-640.

  • 12. Móricz, F. - On the integrability and L1-convergence of double trigonometric series. II, Acta Math. Hungar., 69 (1995), 99-110.

  • 13. Tikhonov, S. - On L1-convergence of Fourier series, J. Math. Anal. Appl., 347 (2008), 416-427.

  • 14. Zhou, S.P. - What condition can correctly generalizes monotonicity in L1 −convergence of sine series?, Science China (Sci. Sin. Math.), 40 (2010), 801-812 (in Chinese).

  • 15. Zygmund, A. - Trigonometric Series, 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959.

1. Bokayev, N.A.; Mukanov, Zh.B. - Weighted integrability of double trigonometric series and of double series with respect to multiplicative systems with coefficients of class R+ 0 BV S2, Translation of Mat. Zametki, 91 (2012), 617-620. Math. Notes, 91 (2012), 575-578.

2. Chen, C.P. - Integrability and L-convergence of multiple trigonometric series, Bull. Austral. Math. Soc., 49 (1994), 333-339.

3. Chen, C.P.; Wu, H.C.; M´oricz, F. - Pointwise convergence of multiple trigono- metric series, J. Math. Anal. Appl., 185 (1994), 629-646.

4. Chen, C.P.; Chauang, Y.W. - L1-convergence of double Fourier series, Chinese J. Math., 19 (1991), 391-410.

5. Kaur, K.; Bhatia, S.S.; Ram, B. - L1-convergence of complex double Fourier series, Proc. Indian Acad. Sci. Math. Sci., 113 (2003), 355-363.

6. Kaur, J.; Bhatia, S.S. - Integrability and L1-convergence of double cosine trigono- metric series, Anal. Theory Appl., 27 (2011), 32-39.

7. Leindler, L. - On the degree of approximation of continuous functions, Acta Math. Hungar., 104 (2004), 105-113.

8. Leindler, L. - Embedding results regarding strong approximation of sine series, Acta. Sci. Math. (Szeged), 71 (2005), 91-103.

9. Leindler, L. - On L1 −convergence of sine series, Anal. Math., 38 (2012), 123-133.

10. Móricz, F. - On the integrability and L1- convergence of double trigonometric series, Studia Math., 98 (1991), 203-225.

11. Móricz, F. - Convergence and integrability of double trigonometric series with coeffi- cients of bounded variation, Proc. Amer. Math. Soc., 102 (1988), 633-640.

12. Móricz, F. - On the integrability and L1-convergence of double trigonometric series. II, Acta Math. Hungar., 69 (1995), 99-110.

13. Tikhonov, S. - On L1-convergence of Fourier series, J. Math. Anal. Appl., 347 (2008), 416-427.

14. Zhou, S.P. - What condition can correctly generalizes monotonicity in L1 −convergence of sine series?, Science China (Sci. Sin. Math.), 40 (2010), 801-812 (in Chinese).

15. Zygmund, A. - Trigonometric Series, 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959.

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