On weighted integrability of double sine series. (English) Zbl 1440.42011
Summary: We introduce a new class of double sequences \(D G M(\alpha, \beta, \gamma, r)\) called double general monotone and some new class of weight functions to study the weighted integrability of double sine series. Some results of D. Yu et al. [Anal. Math. 38, No. 2, 83–104 (2012; Zbl 1265.42022)] are also generalized.
MSC:
42A20 | Convergence and absolute convergence of Fourier and trigonometric series |
42A05 | Trigonometric polynomials, inequalities, extremal problems |
42A32 | Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) |
40A05 | Convergence and divergence of series and sequences |
42B05 | Fourier series and coefficients in several variables |
Keywords:
double sine series; weighted integrability; classes of double sequences; generalized monotonicity; weight functionsCitations:
Zbl 1265.42022References:
[1] | Chen, C. P., Weighted integrability and \(L^1\)-convergence of multiple trigonometric series, Studia Math., 108, 177-190 (1994) · Zbl 0821.42007 |
[2] | Chen, C. P.; Luor, D. C., Two-parameter Hardy-Littlewood inequality and its variants, Studia Math., 139, 9-27 (2000) · Zbl 0967.42015 |
[3] | Duzinkiewicz, K.; Szal, B., On the uniform convergence of double sine series, Colloq. Math., 151, 71-95 (2018) · Zbl 1388.42012 |
[4] | Dyachenko, M.; Tikhonov, S., A Hardy-Littlewood theorem for multiple series, J. Math. Anal. Appl., 339, 503-510 (2008) · Zbl 1283.42031 |
[5] | Dyachenko, M.; Tikhonov, S., General monotone sequences and convergence of trigonometric series, (Topics in Classical Analysis and Applications in Honor of Daniel Waterman (2008), World Sci. Publ.), 88-101 · Zbl 1167.42303 |
[6] | Kórus, P.; Móricz, F., On the uniform convergence of double sine series, Studia Math., 193, 79-97 (2009) · Zbl 1167.42002 |
[7] | Leindler, L., A new extension of monotone sequence and its applications, JIPAM. J. Inequal. Pure Appl. Math., 7, 1, Article 39 pp. (2006) · Zbl 1132.26304 |
[8] | Leindler, L., On the uniform convergence of single and double sine series, Acta Math. Hungar., 140, 232-242 (2013) · Zbl 1299.42007 |
[9] | Marzuq, M. M.H., Integrability theorem of multiple trigonometric series, J. Math. Anal. Appl., 157, 337-345 (1991) · Zbl 0729.42012 |
[10] | Ram, B.; Bhatia, S., On weighted integrability of double cosine series, J. Math. Anal. Appl., 208, 510-519 (1997) · Zbl 0879.42008 |
[11] | Szal, B., A new class of numerical sequences and its applications to uniform convergence of sine series, Math. Nachr., 284, 14-15, 1985-2002 (2011) · Zbl 1230.40004 |
[12] | Szal, B., On L-convergence of trigonometric series, J. Math. Anal. Appl., 373, 449-463 (2011) · Zbl 1204.42010 |
[13] | Tikhonov, S. Yu., On the integrability of the trigonometric series, Math. Notes, 78, 3-4, 437-442 (2005) · Zbl 1136.42003 |
[14] | Tikhonov, S. Yu., Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., 326, 1, 721-735 (2007) · Zbl 1106.42003 |
[15] | Tikhonov, S. Yu., On the uniform convergence of trigonometric series, Mat. Zametki. Mat. Zametki, Math. Notes, 81, 1-2, 268-274 (2007), translation in · Zbl 1183.42005 |
[16] | Tikhonov, S. Yu., Best approximation and moduli of smoothness: computation and equivalence theorems, J. Approx. Theory, 153, 19-39 (2008) · Zbl 1215.42002 |
[17] | Yu, D.; Zhou, P.; Zhou, S., Mean bounded variation condition and applications in double trigonometric series, Anal. Math., 38, 83-104 (2012) · Zbl 1265.42022 |
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