Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

Front Cover
Springer, Jul 8, 2016 - Mathematics - 231 pages
This monograph deals with the mathematics of extending given partial data-sets obtained from experiments; Experimentalists frequently gather spectral data when the observed data is limited, e.g., by the precision of instruments; or by other limiting external factors. Here the limited information is a restriction, and the extensions take the form of full positive definite function on some prescribed group. It is therefore both an art and a science to produce solid conclusions from restricted or limited data.
While the theory of is important in many areas of pure and applied mathematics, it is difficult for students and for the novice to the field, to find accessible presentations which cover all relevant points of view, as well as stressing common ideas and interconnections. We have aimed at filling this gap, and we have stressed hands-on-examples.
 

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Contents

1 Introduction
1
2 Extensions of Continuous Positive Definite Functions
17
3 The Case of More General Groups
47
4 Examples
67
5 Type I vs Type II Extensions
93
6 Spectral Theory for Mercer Operators and Implications for ExtF
115
7 Greens Functions
151
8 Comparing the Different RKHSs HF and HK
170
9 Convolution Products
193
10 Models for and Spectral Representations of OperatorExtensions
196
11 Overview and Open Questions
217
References
219
Index
229
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