INTEGRAL REPRESENTATION OF HYPERBOLICALLY CONVEX FUNCTIONS

Abstract

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 $k(x)$ $\left(x\in \mathbb{R}_0^{\infty}\right)$