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Uniform convergence of double trigonometric series

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EN
It is shown that under certain conditions on ${c_{jk}}$, the rectangular partial sums $s_{mn}(x,y)$ converge uniformly on $T^2$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is $∑_{|k|= n}^∞ |Δc_k| = o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $nc_{n} = o(1)$ as n → ∞. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].
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Twórcy
  • Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China, cpchen@math.nthu.edu.tw
autor
  • Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
Bibliografia
  • [CJ] T. W. Chaundy and A. E. Jolliffe, The uniform convergence of a certain class of trigonometric series, Proc. London Math. Soc. (2) 15 (1916), 214-216.
  • [C1] C.-P. Chen, Weighted integrability and $L^1$-convergence of multiple trigonometric series, Studia Math. 108 (1994), 177-190.
  • [C2] C.-P. Chen, Integrability of multiple trigonometric series and Parseval's formula, J. Math. Anal. Appl. 186 (1994), 182-199.
  • [CL] C.-P. Chen and C.-C. Lin, Integrability, mean convergence, and Parseval's formula for double trigonometric series, preprint.
  • [D] M. I. Dyachenko, The rate of u-convergence of multiple Fourier series, Acta Math. Hungar. 68 (1995), 55-70.
  • [J] A. E. Jolliffe, On certain trigonometric series which have a necessary and sufficient condition for uniform convergence, Math. Proc. Cambridge Philos. Soc. 19 (1921), 191-195.
  • [K] J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53.
  • [M1] F. Móricz, Convergence and integrability of double trigonometric series with coefficients of bounded variation, Proc. Amer. Math. Soc. 102 (1988), 633-640.
  • [M2] F. Móricz, On the integrability and $L^1$-convergence of double trigonometric series, Studia Math. 98 (1991), 203-225.
  • [M3] F. Móricz, On the integrability of double cosine and sine series I, J. Math. Anal. Appl. 154 (1991), 452-465.
  • [N] J. R. Nurcombe, On the uniform convergence of sine series with quasimonotone coefficients, ibid. 166 (1992), 577-581.
  • [S] O. Szász, Quasi-monotone series, Amer. J. Math. 70 (1948), 203-206.
  • [XZ] T. F. Xie and S. P. Zhou, The uniform convergence of certain trigonometric series, J. Math. Anal. Appl. 181 (1994), 171-180.
  • [Z] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, 1968.
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Bibliografia
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