INTEGRAL REPRESENTATION OF EVEN POSITIVE DEFINITE BOUNDED FUNCTIONS OF AN INFINITE NUMBER OF VARIABLES

Abstract

In this article the integral representation for bounded even positive functions $k(x)$\linebreak $\left(x\in \mathbb{R}^\infty=\mathbb{R}\times\mathbb{R}\times\dots \right)$ is proved. We understand the positive the positive definite in the integral sense with integration respects to measure $d\theta(x)= p(x_1)dx_1\otimes p(x_2)dx_2\otimes \dots$\linebreak $\left(p(x)=\sqrt{\frac{1}{\pi}}e^{-x^2} \right)$. This integral representation has the form \begin{equation}\label{ovl1.0} k(x)=\int\limits_{l_2^+} {\rm Cos}\,\lambda_ix_id\rho(\lambda) \end{equation} Equality stands to reason for almost all $x\in \mathbb{R}^\infty$. $l_2^+$ space consists of those vectors $\lambda\in\mathbb{R}^\infty_+=\mathbb{R}^1_+\times \mathbb{R}^1_+\times\dots\left| \sum\limits_{i=1}^\infty \lambda_i^2 <\infty\right.$. Conversely, every integral of form~\eqref{ovl1.0} is bounded by even positively definite function $k(x)$ $x\in\mathbb{R}^\infty$. As a result, from this theorem we shall get generalization of theorem of R.~A.~Minlos--V.~V.~Sazonov \cite{lov2,lov3} in case of bounded even positively definite functions $k(x)$ $(x\in H)$, which are continuous in $O$ in $j$"=topology.