Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

Description

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. Front Matter....Pages i-xxvi Introduction....Pages 1-16 Extensions of Continuous Positive Definite Functions....Pages 17-46 The Case of More General Groups....Pages 47-66 Examples....Pages 67-92 Type I vs. Type II Extensions....Pages 93-113 Spectral Theory for Mercer Operators, and Implications for \(Ext\left (F\right )\) ....Pages 115-150 Green’s Functions....Pages 151-169 Comparing the Different RKHSs \(\mathcal{H}_{F}\) and \(\mathcal{H}_{K}\) ....Pages 171-191 Convolution Products....Pages 193-195 Models for, and Spectral Representations of, Operator Extensions....Pages 197-216 Overview and Open Questions....Pages 217-218 Back Matter....Pages 219-233