Book: Positive Definite and Definitizable Functions
Zoltán Sasvári
Positive Definite and Definitizable Functions.
Berlin: Akademie Verlag, 1994.
Errata (Version September 19, 2019)
A few copies are still available from the author.
Contents
1 Positive definite functions
1.1 Notation and preliminaries
1.2 Inner product spaces defined by complex-valued functions
1.3 Construction of the Hilbert space H(f)
1.4 Some inequalities
1.5 Convergence of positive definite functions
1.6 Convolution and positive definite functions
1.7 Integrally positive definite functions
1.8 Extremal positive definite functions and irreducible representations
1.9 Bochner's theorem
1.10 Examples of positive definite functions
2 Some applications of positive definite functions
2.1 Duality of abelian groups
2.2 Spectral decomposition of unitary representations of abelian groups
2.3 Unitary representations of compact groups
3 Decomposition of positive definite functions
3.1 Decomposition of measurable positive definite functions
3.2 Decomposition of arbitrary positive definite functions
3.3 A finer decomposition
3.4 Equivalence of local and global measurability
4 Extension of positive definite functions
4.1 Extension from symmetric sets
4.2 The one dimensional case
4.3 The multidimensional case
4.4 Extension from subgroups
5 Functions with a finite number of negative squares
5.1 Connection with unitary representations in Pontryagin spaces
5.2 Characterization of bounded functions with k negative squares
5.3 Measurable functions with k negative squares
5.4 Functions of finite rank
5.5 Definitizability of functions with k negative squares
6 Definitizable functions
6.1 Definition and some simple properties
6.2 The singularities of definitizable functions
6.3 Functions of finite order
6.4 Integral representation of functions f Î Pc(γ,k) with γ Î Γ
6.5 Further results
6.6 The singularities of a function with k negative squares
A Pontryagin spaces
B Invariant subspaces of unitary operators in Pontryagin spaces
Bibliography
List of symbols