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Monotone Matrix Functions and Analytic Continuation

  • Book
  • © 1974

Overview

Authors:
  1. William F. Donoghue

    1. University of California, Irvine, USA

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 207)

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Table of contents (19 chapters)

  1. Front Matter

    Pages i-ix
    Download chapter PDF

  2. Preliminaries

    • William F. Donoghue Jr.
    Pages 1-17

  3. Pick Functions

    • William F. Donoghue Jr.
    Pages 18-33

  4. Pick Matrices and Loewner Determinants

    • William F. Donoghue Jr.
    Pages 34-41

  5. Fatou Theorems

    • William F. Donoghue Jr.
    Pages 42-49

  6. The Spectral Theorem

    • William F. Donoghue Jr.
    Pages 50-62

  7. One-Dimensional Perturbations

    • William F. Donoghue Jr.
    Pages 63-66

  8. Monotone Matrix Functions

    • William F. Donoghue Jr.
    Pages 67-77

  9. Sufficient Conditions

    • William F. Donoghue Jr.
    Pages 78-84

  10. Loewner’s Theorem

    • William F. Donoghue Jr.
    Pages 85-87

  11. Reproducing Kernels

    • William F. Donoghue Jr.
    Pages 88-93

  12. Nagy-Koranyi Proof of Loewner’s Theorem

    • William F. Donoghue Jr.
    Pages 94-99

  13. The Cauchy Interpolation Problem

    • William F. Donoghue Jr.
    Pages 100-116

  14. Interpolation by Pick Functions

    • William F. Donoghue Jr.
    Pages 117-127

  15. The Interpolation of Monotone Matrix Functions

    • William F. Donoghue Jr.
    Pages 128-133

  16. Almost Positive Matrices

    • William F. Donoghue Jr.
    Pages 134-139

  17. The Analytic Continuation of Bergman Kernels

    • William F. Donoghue Jr.
    Pages 140-145

  18. The Loewner-FitzGerald Theorem

    • William F. Donoghue Jr.
    Pages 146-153

  19. Loewner’s Differential Equation

    • William F. Donoghue Jr.
    Pages 154-164

  20. More Analytic Continuation

    • William F. Donoghue Jr.
    Pages 165-169
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Keywords

About this book

A Pick function is a function that is analytic in the upper half-plane with positive imaginary part. In the first part of this book we try to give a readable account of this class of functions as well as one of the standard proofs of the spectral theorem based on properties of this class. In the remainder of the book we treat a closely related topic: Loewner's theory of monotone matrix functions and his analytic continuation theorem which guarantees that a real function on an interval of the real axis which is a monotone matrix function of arbitrarily high order is the restriction to that interval of a Pick function. In recent years this theorem has been complemented by the Loewner-FitzGerald theorem, giving necessary and sufficient conditions that the continuation provided by Loewner's theorem be univalent. In order that our presentation should be as complete and trans­ parent as possible, we have adjoined short chapters containing the in­ formation about reproducing kernels, almost positive matrices and certain classes of conformal mappings needed for our proofs.

Authors and Affiliations

  • University of California, Irvine, USA

    William F. Donoghue

Bibliographic Information

  • Book Title: Monotone Matrix Functions and Analytic Continuation

  • Authors: William F. Donoghue

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-642-65755-9

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1974

  • Softcover ISBN: 978-3-642-65757-3Published: 23 December 2011

  • eBook ISBN: 978-3-642-65755-9Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: X, 184

  • Topics: Analysis

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