POSITIVE DEFINITE FUNCTIONS ON INFINITE-DIMENSIONAL CONVEX CONES by Helge Glockner ( 9780821832561 )

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Title:

POSITIVE DEFINITE FUNCTIONS ON INFINITE-DIMENSIONAL CONVEX CONES

By:

Format:

Paperback

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ISBN 10:

0821832565

ISBN 13:

9780821832561

Publisher:

AMERICAN MATHEMATICAL SOCIETY

Pub. date:

15 September, 2003

Edition:

New ed.

Series:

Memoirs of the American Mathematical Society v. 166

Pages:

128

Description:

A memoir that studies positive definite functions on convex subsets of finite- or infinite-dimensional vector spaces. It studies representations of convex cones by positive operators on Hilbert spaces. It also studies the interplay between positive definite functions and representations of convex cones.

Synopsis:

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.

Publication:

Imprint:

American Mathematical Society

Returns:

Returnable

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