eBook ISBN:  9781470403874 
Product Code:  MEMO/166/789.E 
List Price:  $62.00 
MAA Member Price:  $55.80 
AMS Member Price:  $37.20 
eBook ISBN:  9781470403874 
Product Code:  MEMO/166/789.E 
List Price:  $62.00 
MAA Member Price:  $55.80 
AMS Member Price:  $37.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 166; 2003; 128 ppMSC: Primary 43; Secondary 20; 44; 46
This memoir is devoted to the study of positive definite functions on convex subsets of finite or infinitedimensional 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 nonempty 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 operatorvalued 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 vectorvalued 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 infinitedimensional 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 nonempty 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 operatorvalued 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 vectorvalued 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.
ReadershipGraduate students and research mathematicians interested in analysis.

Table of Contents

Chapters

I. Preliminaries and preparatory results

1. Bounded and unbounded operators

2. Conevalued measures

3. Measures on topological spaces

4. Projective limits of conevalued measures

5. Holomorphic functions

6. Involutive semigroups and their representations

7. Positive definite kernels and functions

8. $C$*algebras associated with involutive semigroups

9. Integral representations of positive definite functions

10. Convex cones and their faces

11. Examples of convex cones

12. Conelike semigroups: definition and examples

13. Representations of conelike semigroups I

14. Fourier and Laplace transforms

15. Generalized Bochner and Stone theorems

II. Main results

16. Nussbaum Theorem for open convex cones

17. Positive definite functions on convex cones with nonempty interior

18. Positive definite functions on convex sets

19. Associated Hilbert spaces and representations

20. Nussbaum Theorem for generating convex cones

21. Representations of conelike semigroups II

22. Associated unitary representations

23. Holomorphic extension of unitary representations

24. Holomorphic extension of representations of nuclear groups


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This memoir is devoted to the study of positive definite functions on convex subsets of finite or infinitedimensional 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 nonempty 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 operatorvalued 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 vectorvalued 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 infinitedimensional 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 nonempty 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 operatorvalued 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 vectorvalued 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.
Graduate students and research mathematicians interested in analysis.

Chapters

I. Preliminaries and preparatory results

1. Bounded and unbounded operators

2. Conevalued measures

3. Measures on topological spaces

4. Projective limits of conevalued measures

5. Holomorphic functions

6. Involutive semigroups and their representations

7. Positive definite kernels and functions

8. $C$*algebras associated with involutive semigroups

9. Integral representations of positive definite functions

10. Convex cones and their faces

11. Examples of convex cones

12. Conelike semigroups: definition and examples

13. Representations of conelike semigroups I

14. Fourier and Laplace transforms

15. Generalized Bochner and Stone theorems

II. Main results

16. Nussbaum Theorem for open convex cones

17. Positive definite functions on convex cones with nonempty interior

18. Positive definite functions on convex sets

19. Associated Hilbert spaces and representations

20. Nussbaum Theorem for generating convex cones

21. Representations of conelike semigroups II

22. Associated unitary representations

23. Holomorphic extension of unitary representations

24. Holomorphic extension of representations of nuclear groups