This memoir is devoted to the study of positive definite
functions on convex subsets of finite- or infinite-dimensional vector spaces,
and to the study of representations of convex cones by positive operators on
Hilbert spaces. Given a convex subset $\Omega\subseteq V$ of a real vector
space $V$, we show that a function
$\phi\!:\Omega\to\mathbb{R}$ is the Laplace transform of a positive
measure $\mu$ on the algebraic dual space $V^*$ if and only
if $\phi$ is continuous along line segments and positive definite. If
$V$ is a topological vector space and $\Omega\subseteq V$ an open
convex cone, or a convex cone with non-empty interior, we describe sufficient
conditions for the existence of a representing measure $\mu$ for
$\phi$ on the topological dual space$V'$. The results are
used to explore continuity properties of positive definite functions on convex
cones, and their holomorphic extendibility to positive definite functions on
the associated tubes $\Omega+iV\subseteq V_{\mathbb{C}}$. We also study the
interplay between positive definite functions and representations of convex
cones, and derive various characterizations of those representations of convex
cones on Hilbert spaces which are Laplace transforms of spectral measures.
Furthermore, for scalar- or operator-valued positive definite functions which
are Laplace transforms, we realize the associated reproducing kernel Hilbert
space as an $L^2$-space $L^2(V^*,\mu)$ of vector-valued
functions and link the natural translation operators on the reproducing kernel
space to multiplication operators on $L^2(V^*,\mu)$, which gives us
refined information concerning the norms of these operators.This memoir is
devoted to the study of positive definite functions on convex subsets of
finite- or infinite-dimensional vector spaces, and to the study of
representations of convex cones by positive operators on Hilbert spaces. Given
a convex subset $\Omega\subseteq V$ of a real vector space $V$, we
show that a function $\phi\!:\Omega\to\mathbb{R}$ is the Laplace
transform of a positive measure $\mu$ on the algebraic dual space
$V^*$ if and only if $\phi$ is continuous along line segments
and positive definite. If $V$ is a topological vector space and
$\Omega\subseteq V$ an open convex cone, or a convex cone with non-empty
interior, we describe sufficient conditions for the existence of a representing
measure $\mu$ for $\phi$ on the topological dual space
$V'$. The results are used to explore continuity properties of
positive definite functions on convex cones, and their holomorphic
extendibility to positive definite functions on the associated tubes
$\Omega+iV\subseteq V_\mathbb C$. We also study the interplay between
positive definite functions and representations of convex cones, and derive
various characterizations of those representations of convex cones on Hilbert
spaces which are Laplace transforms of spectral measures. Furthermore, for
scalar- or operator-valued positive definite functions which are Laplace
transforms, we realize the associated reproducing kernel Hilbert space as an
$L^2$-space $L^2(V^*,\mu)$ of vector-valued functions and
link the natural translation operators on the reproducing kernel space to
multiplication operators on $L^2(V^*,\mu)$, which gives us refined
information concerning the norms of these operators.

Readership

Graduate students and research mathematicians interested in
analysis.