Weak convergence of vector measures on topological spaces and its applications

拓扑空间矢量测度的弱收敛及其应用

• 批准号: 09640173
• 负责人: KAWABE Jun
• 金额: $ 1.15万
• 依托单位: SHINSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640173/
• 关键词: tensor product of vector measures; weak convergence of vec-tor measures; fuzzy optimal control; Norlund summability; rearrangement of vector-valued functions; digital control of continuous-time linear systems; martingale; Kahlerian submersion; フーリ

We have studied weak convergence of vector measures with values in Banach spaces and nuclear spaces, and have applied it to several interesting problems in real analysis, probability theory, control theory and so on. Some of our important results are as follows :1. We proved that the weak convergence of a net of tensor products of vector measures with values in nuclear spaces follows from that of the corresponding net of real product measures.2. We gave weak (sequential) compactness criteria for Radon vector measures with values in several locally convex spaces. Consequently, our result contains previous criteria obtained by Prokhorov-LeCam for real measures and by Marz and Shortt for Banach space valued measures.3. We extended a type of Strassen's theorem for probability measures to positive vector measures with values in the weak dual of a barreled locally convex space which has certain order conditions.4. We examined the convergence of feedback laws constructed by Mamdani method and the convergence of solutions of a non-linear state equation in fuzzy control system. As an application, we ensured the existence of fuzzy optimal control.5. We proved a theorem on the generalized Norlund summability (N, pn, qn) of Legendre series, which contains previous results by V.Singh, N.P.Mishra and Ajit Singh. It is also an Legendre analogue of a result due to Rajagopal.6. We proved some theorems on the generalized Norlund summability (N, pn, qn) of Fourier series and its conjugate series, which contain several previous results by A.N.Singh and S.P.Khare.

本文研究了取值于Banach空间和核空间的向量测度的弱收敛性，并将其应用于真实的分析、概率论、控制论等领域中的一些有趣的问题，主要结果如下：1.证明了向量测度张量积网的弱收敛性可以从相应的真实的乘积测度网的弱收敛性得到.给出了取值于几个局部凸空间的Radon向量测度的弱（序列）紧性准则。因此，我们的结果包含了Prokhorov-LeCam关于真实的测度和Marz和Shortt关于Banach空间值测度的判别准则.将一类关于概率测度的斯特拉森定理推广到具有一定序条件的桶形局部凸空间的弱对偶中的正向量测度.研究了用Mamdani方法构造的反馈律的收敛性和模糊控制系统非线性状态方程解的收敛性。作为应用，证明了模糊最优控制的存在性.证明了Legendre级数的广义Norlund可和性（N，pn，qn）的一个定理，它包含了V.Singh，N. P. Mishra和Ajit Singh的结果.这也是一个勒让德模拟的结果，由于Rajagopal。6.证明了Fourier级数及其共轭级数的广义Norlund可和性（N，pn，qn）的几个定理，这些定理包含了A. N. Singh和S. P. Khare的几个结果.

项目成果

Y.Okuyama and I.Miyamoto: "On the generalized Norlund summability of Fourier series and its conjugate series" Far East J.Math.Sci.6. 561-574 (1998)

Y.Okuyama 和 I.Miyamoto：“论傅里叶级数及其共轭级数的广义 Norlund 可求性”远东 J.Math.Sci.6。

J.Kawabe: "Weak convergence of tensor products of vector mea-sures with values in nuclear spaces" Bull.Austral.Math.Soc.59. 451-460 (1999)

J.Kawabe：“矢量测量的张量积与核空间中的值的弱收敛”Bull.Austral.Math.Soc.59。

J.Kawabe: "Compactness criteria for the weak topology of vector measures" Proc.Sem.Banach spaces Rel.Topics. 2. 15-18 (1997)

J.Kawabe：“矢量测度弱拓扑的紧致性标准”Proc.Sem.Banach 空间 Rel.Topics。

J.Kawabe: "Weak convergence of tensor products of vector measures in nuclear spaces" Proceedings of China and Japan Joint Symposium on Applied Functional Analysis. (掲載決定).

J.Kawabe：“核空间矢量测度张量积的弱收敛”中日应用泛函分析联合研讨会论文集（已出版）。

河邊 淳: "ベクトル測度のテンソル積の弱収束" 数理解析研究所講究録. (掲載決定).

Jun Kawabe：“矢量测量的张量积的弱收敛”数学分析研究所的 Kokyuroku（已决定出版）。