Generalized functions in infinite dimensional analysis.

*(English)*Zbl 0929.46031The present article examines an approach to infinite-dimensional non-Gaussian analysis and some results of Gaussian analysis are also pointed out. The authors use biorthogonal Appell systems \(\mathbb{A}^\mu\), where \(\mu\) a measure, as pair \((\mathbb{P}^\mu, \mathbb{Q}^\mu)\) of Appell polynomial \(\mathbb{P}^\mu\) and a canonical generalized function \(\mathbb{Q}^\mu\), for defining generalized functions in infinite-dimensional analysis. The article also describes the two measures, such as analytic measure and nondegenerate measure, their properties are identical to the test function spaces and its dual in the nuclear spaces. The representation theorem for distributions is proved. A brief description for characterization of singularity in distributions and their canonical representations is given.

This article also describes the \(S_\mu\)-transform and \(C_\mu\)-transform convolution and shows how the transforms differ in Gaussian and non-Gaussian analysis, by example. Using these transforms, a characterization theorem and Wick product is proved. A brief description of positive distributions is given. This paper generalizes the non-Gaussian analysis to measures which possess more singular logarithmic derivatives. Distributions in Gaussian and infinite non-Gaussian analysis are explained. The authors also assert to handle measures of Poisson type.

This article also describes the \(S_\mu\)-transform and \(C_\mu\)-transform convolution and shows how the transforms differ in Gaussian and non-Gaussian analysis, by example. Using these transforms, a characterization theorem and Wick product is proved. A brief description of positive distributions is given. This paper generalizes the non-Gaussian analysis to measures which possess more singular logarithmic derivatives. Distributions in Gaussian and infinite non-Gaussian analysis are explained. The authors also assert to handle measures of Poisson type.

Reviewer: P.K.Banerji (Jodhpur)

##### MSC:

46F25 | Distributions on infinite-dimensional spaces |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |