On weighted Hilbert spaces and integration of functions of infinitely many variables

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• 作者：Michael Gnewuch^a ; ^{m.gnewuch@unsw.edu.au" class="auth_mail} ; Sebastian Mayer^b ; ^{mayer@ins.uni-bonn.de" class="auth_mail} ; Klaus Ritter^c ; ^{ritter@mathematik.uni-kl.de" class="auth_mail}
• 关键词：Reproducing kernel Hilbert space ; Functions of infinitely many variables ; Tensor product ; Weighted superposition ; Integration problem
• 刊名：Journal of Complexity
• 出版年：April, 2014
• 年：2014
• 卷：30
• 期：2
• 页码：29-47
• 全文大小：422 K

文摘

We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables . The setting is based on a reproducing kernel for functions on , a family of non-negative weights , where varies over all finite subsets of , and a probability measure on . We consider the weighted superposition of finite tensor products of . Under mild assumptions we show that is a reproducing kernel on a properly chosen domain in the sequence space , and that the reproducing kernel Hilbert space is the orthogonal sum of the spaces . Integration on can be defined in two ways, via a canonical representer or with respect to the product measure on . We relate both approaches and provide sufficient conditions for the two approaches to coincide.