About this ebook
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 $\Omega\sub V$ 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 $\Omega\sub V$ 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 $\Omega+iV\sub V_{\mathbb{C}}$. 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 $\Omega\sub V$ of a real vector space $V$, we s...
Discover more
Rate this ebook
Tell us what you think.
Reading information
Smartphones and tablets
Install the Google Play Books app for Android and iPad/iPhone. It syncs automatically with your account and allows you to read online or offline wherever you are.
Laptops and computers
You can listen to audiobooks purchased on Google Play using your computer's web browser.
eReaders and other devices
To read on e-ink devices like Kobo eReaders, you'll need to download a file and transfer it to your device. Follow the detailed Help Center instructions to transfer the files to supported eReaders.
