INTEGRAL REPRESENTATION OF HYPERBOLICALLY CONVEX FUNCTIONS

An article consists of two parts. In the first part the sufficient and necessary conditions for an integral representation of hyperbolically convex (h.c.) functions $k(x)$ $\left(x\in \mathbb{R}^{\infty}= \mathbb{R}^1\times\mathbb{R}^1\times \dots\right)$ are proved. For this purpose in $\mathbb{R}^{\infty}$ we introduce measures $\omega_1(x)$, $\omega_{\frac{1}{2}}(x)$. The positive definiteness of a function will be understood on the integral sense with respect to the measure $\omega_1(x)$. Then we proved that the measure $\rho(\lambda)$ in the integral representation is concentrated on $l_2^+=\bigg\{\lambda \in \mathbb{R}_+^{\infty}= \mathbb{R}_+^1\times\mathbb{R}_+^1\times \dots\Big|\sum\limits_{n=1}^{\infty}\lambda_n^2<\infty\bigg\}$. The equality for $k(x)$ $\left(x\in\mathbb{R}^{\infty} \right)$ is regarded as an equality for almost all $x\in\mathbb{R}^{\infty}$ with respect to measure $\omega_{\frac{1}{2}}(x)$. In the second part we proved the sufficient and necessary conditions for integral representation of h.c. functions $k(x)$ $\big(x\in \mathbb{R}_0^{\infty}$ $\mathrm{~is~a~nuclear~space}\big)$. The positive definiteness of a function $k(x)$ will be understood on the pointwise sense. For this purpose we shall construct a rigging (chain) $\mathbb{R}_0^{\infty}\subset l_2 \subset \mathbb{R}^{\infty}$. Then, given that the projection and inductive topologies are coinciding, we shall obtaine the integral representation for $k(x)$ $\left(x\in \mathbb{R}_0^{\infty}\right)$