Extensions of Positive Definite Functions : Applications and Their Harmonic Analysis / by Palle Jorgensen, Steen Pedersen, Feng Tian.

Format:

Book

Author/Creator:

Jorgensen, Palle., Author.

Pedersen, Steen., Author.

Tian, Feng, Author.

Series:

Lecture Notes in Mathematics, 0075-8434 ; 2160

Language:

English

Subjects (All):

Harmonic analysis.

Topological groups.

Lie groups.

Fourier analysis.

Functional analysis.

Mathematical physics.

Probabilities.

Abstract Harmonic Analysis.

Topological Groups, Lie Groups.

Fourier Analysis.

Functional Analysis.

Mathematical Physics.

Probability Theory and Stochastic Processes.

Local Subjects:

Abstract Harmonic Analysis.

Topological Groups, Lie Groups.

Fourier Analysis.

Functional Analysis.

Mathematical Physics.

Probability Theory and Stochastic Processes.

Physical Description:

1 online resource (XXVI, 231 p. 48 illus., 9 illus. in color.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Summary:

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.

Contents:

Intro

Foreword

Preface

Acknowledgments

Contents

List of Figures

List of Tables

Symbols

1 Introduction

1.1 Two Extension Problems

1.1.1 Where to Find It

1.2 Quantum Physics

1.3 Stochastic Processes

1.3.1 Early Roots

1.3.2 An Application of Lemma 1.1: A Positive Definite Function on an Infinite Dimensional Vector Space

1.4 Overview of Applications of RKHSs

1.4.1 Connections to Gaussian Processes

1.5 Earlier Papers

1.6 Organization

2 Extensions of Continuous Positive Definite Functions

2.1 The RKHS HF

2.1.1 An Isometry

2.2 The Skew-Hermitian Operator D(F) in HF

2.2.1 The Case of Conjugations

2.2.2 Illustration: G=R, Correspondence Between the Two Extension Problems

2.3 Enlarging the Hilbert Space

2.4 Ext1(F) and Ext2(F)

2.4.1 The Case of n=1

2.4.2 Comparison of p.d. Kernels

2.5 Spectral Theory of D(F) and Its Extensions

3 The Case of More General Groups

3.1 Locally Compact Abelian Groups

3.2 Lie Groups

3.2.1 The GNS Construction

3.2.2 Local Representations

3.2.3 The Convex Operation in Ext(F)

4 Examples

4.1 The Case of G=Rn

4.2 The Case of G=R/Z

4.3 Example: ei2πx

4.4 Example: e-|x| in (-a,a), Extensions to T=R/Z

4.4.1 General Consideration

4.5 Example: e-|x| in (-a,a), Extensions to R

4.6 Example: A Non-extendable p.d. Function in a Neighborhood of Zero in G=R2

4.6.1 A Locally Defined p.d. Functions F on G=R2 with Ext(F)=.

5 Type I vs. Type II Extensions

5.1 Pólya Extensions

5.2 Main Theorems

5.2.1 Some Applications

5.3 The Deficiency-Indices of D(F)

5.3.1 Pólya-Extensions

5.4 The Example 5.3, Green's Function, and an HF-ONB

6 Spectral Theory for Mercer Operators, and Implications for Ext(F)

6.1 Groups, Boundary Representations, and Renormalization

6.2 Shannon Sampling, and Bessel Frames.

6.3 Application: The Case of F2 and Rank-1 Perturbations

6.4 Positive Definite Functions, Green's Functions, and Boundary

6.4.1 Connection to the Energy Form Hilbert Spaces

7 Green's Functions

7.1 The RKHSs for the Two Examples F2 and F3 in Table 5.1

7.1.1 Green's Functions

7.1.1.1 Summary: Conclusions for the Two Examples

7.1.2 The Case of F2(x)=1-|x|, x(-12,12)

7.1.2.1 Pinned Brownian Motion

7.1.3 The Case of F3(x)=e-|x|, x(-1,1)

7.1.4 Integral Kernels and Positive Definite Functions

7.1.5 The Ornstein-Uhlenbeck Process Revisited

7.1.6 An Overview of the Two Cases: F2 and F3.

7.2 Higher Dimensions

8 Comparing the Different RKHSs HF and HK

8.1 Applications

8.2 Radially Symmetric Positive Definite Functions

8.3 Connecting F and F When F Is a Positive Definite Function

8.4 The Imaginary Part of a Positive Definite Function

8.4.1 Connections to, and Applications of, Bochner's Theorem

9 Convolution Products

10 Models for, and Spectral Representations of, OperatorExtensions

10.1 Model for Restrictions of Continuous p.d. Functions on R

10.2 A Model of ALL Deficiency Index-(1,1) Operators

10.2.1 Momentum Operators in L2(0,1)

10.2.2 Restriction Operators

10.3 The Case of Indices (d,d) Where d>1

10.4 Spectral Representation of Index (1,1) Hermitian Operators

11 Overview and Open Questions

11.1 From Restriction Operator to Restriction of p.d. Function

11.2 The Splitting HF=HF(atom)HF(ac) HF(sing)

11.3 The Case of G=R1

11.4 The Extreme Points of Ext(F) and { F}

References

Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-39780-X

OCLC:

953574514