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  • © 2016

Extensions of Positive Definite Functions

Applications and Their Harmonic Analysis

  • Makes connections between disparate areas of mathematics and its applications

  • Student friendly features: Includes worked examples, tables, images, and graphs

  • Features a mix of modern and classical analysis: The exposition combines novel approaches and new research advances with classical core areas of mathematics

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2160)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xxvi
  2. Introduction

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 1-16
  3. Extensions of Continuous Positive Definite Functions

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 17-46
  4. The Case of More General Groups

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 47-66
  5. Examples

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 67-92
  6. Type I vs. Type II Extensions

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 93-113
  7. Spectral Theory for Mercer Operators, and Implications for \(Ext\left (F\right )\)

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 115-150
  8. Green’s Functions

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 151-169
  9. Comparing the Different RKHSs \(\mathcal{H}_{F}\) and \(\mathcal{H}_{K}\)

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 171-191
  10. Convolution Products

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 193-195
  11. Models for, and Spectral Representations of, Operator Extensions

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 197-216
  12. Overview and Open Questions

    • Palle Jorgensen, Steen Pedersen, Feng Tian
    Pages 217-218
  13. Back Matter

    Pages 219-233

About this book

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.

Keywords

  • 47L60,46N30,46N50,42C15,65R10.
  • positive definite functions
  • unitary representations
  • spectral theory
  • Pontryagin-Bochner duality
  • reproducing kernel Hilbert space
  • completely monotone functions
  • Gaussian processes
  • Gelfand-triple
  • Unbounded operators
  • von Neumann’s theory of deficiency indices

Authors and Affiliations

  • Department of Mathematics, The University of Iowa , Iowa City, USA

    Palle Jorgensen

  • Department of Mathematics, Wright State University , Dayton, USA

    Steen Pedersen

  • Department of Mathematics, Hampton University , Hampton, USA

    Feng Tian

Bibliographic Information

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions