White noise calculus and Fock space
In the beginning, Hida had started his investigations on white noise calculus with strong interest in infinite-dimensional rotation groups and related functional analysis. The author stands on this view point and gives us a bird's eye view of white noise calculus. Let \(E \subset H \subset E'\) be a Gel'fand triplet. The group \(O(E;H) = \{g \in \text{GL} (E)\); \(| g \xi | = | \xi |\), \(\forall \xi \in E\}\) is called the infinite-dimensional rotational group of this triplet. An operator which commutes with this rotation group is called a Laplacian. There exist three well-known Laplacians, Gross Laplacian \(\Delta_ G\), its dual operator \(\Delta_ G^*\) and the number operator \(N\). Then, \(\{\Xi \in {\mathcal L} ((E), (E)^*)\); Laplacian\} is generated by \(\Delta_ G^*\), \(\Delta_ G\) and \(N\), and \(\{\Xi \in {\mathcal L} ((E), (E))\); Laplacian\} is generated by \(\Delta_ G\) and \(N\). Here \((E)\) is the space of testing functionals and \((E)^*\) is the space of Hida distributions. The infinite-dimensional Fourier transformation is introduced and some of its properties are shown.
harmonic analysis of infinitely many variables
scientific article; zbMATH DE number 595993