本文讨论广义函数Denjoy积分的收敛性问题. 首先给出了广义Denjoy可积函数空间中强收敛、弱收敛、弱* 收敛和广义函数Denjoy积分收敛的关系; 证明拟一致收敛是广义函数Denjoy积分 收敛的一个充分必要条件; 最后指出了Denjoy可积广义函数列弱* 收敛与强收敛等价当且仅当原函数等度连续.
Abstract
This paper discusses some problems on the convergence of distributional Denjoy integral. Firstly, we give the relations between strong convergence, weak convergence, weak* convergence and distributional Denjoy integral convergence in the space of Denjoy integrable distributions. We prove the quasi-uniform convergence is a sufficient and necessary condition of the convergence of distributional Denjoy integral. Finally, we show that weak* convergence and strong convergence are equivalent if and only if the sequence of the primitive of Denjoy integrable distribution are equicontinuous.
关键词
广义函数Denjoy积分 /
强收敛 /
弱收敛
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Key words
Denjoy distributional integral /
strong convergence /
weak convergence
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参考文献
[1] Erik T., The Denjoy distributional integration, Real Anal. Exchang, 2008, 33: 51-82.
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[5] Schwabik S., Ye G. J., Topics In Banach Space Integration, Singapore: World Scientific, 2005.
[6] Zhong C. K., Fan X. L., Chen W. Y., Introduction to Nonlinear Functional Analysis, Lanzhou: Lanzhou University Press, 1998 (in Chinese).
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