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Operator Relations Characterizing Derivatives

Authors: König, Hermann, Milman, Vitali

  • Develops an operator viewpoint for functional equations in classical function spaces of analysis
  • Demonstrates the rich, operator-type structure behind the fundamental notion of the derivative and its relatives in analysis
  • Fills a gap in mathematical literature; it explores algebraic properties of the derivative in a purely analytic setup
  • Gives a self-contained presentation; most proofs are written in detail
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eBook $79.99
price for USA in USD (gross)
  • ISBN 978-3-030-00241-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.99
price for USA in USD
  • ISBN 978-3-030-00240-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

This monograph develops an operator viewpoint for functional equations in classical function spaces of analysis, thus filling a void in the mathematical literature. Major constructions or operations in analysis are often characterized by some elementary properties, relations or equations which they satisfy. The authors present recent results on the problem to what extent the derivative is characterized by equations such as the Leibniz rule or the Chain rule operator equation in Ck-spaces. By localization, these operator equations turn into specific functional equations which the authors then solve. The second derivative, Sturm-Liouville operators and the Laplacian motivate the study of certain "second-order" operator equations. Additionally, the authors determine the general solution of these operator equations under weak assumptions of non-degeneration. In their approach, operators are not required to be linear, and the authors also try to avoid continuity conditions. The Leibniz rule, the Chain rule and its extensions turn out to be stable under perturbations and relaxations of assumptions on the form of the operators. The results yield an algebraic understanding of first- and second-order differential operators. Because the authors have chosen to characterize the derivative by algebraic relations, the rich operator-type structure behind the fundamental notion of the derivative and its relatives in analysis is discovered and explored.

The book does not require any specific knowledge of functional equations. All needed results are presented and proven and the book is addressed to a general mathematical audience.


Table of contents (9 chapters)

  • Introduction

    König, Hermann (et al.)

    Pages 1-8

  • Regular Solutions of Some Functional Equations

    König, Hermann (et al.)

    Pages 9-28

  • The Leibniz Rule

    König, Hermann (et al.)

    Pages 29-52

  • The Chain Rule

    König, Hermann (et al.)

    Pages 53-73

  • Stability and Rigidity of the Leibniz and the Chain Rules

    König, Hermann (et al.)

    Pages 75-90

Buy this book

eBook $79.99
price for USA in USD (gross)
  • ISBN 978-3-030-00241-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.99
price for USA in USD
  • ISBN 978-3-030-00240-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Operator Relations Characterizing Derivatives
Authors
Copyright
2018
Publisher
Birkhäuser Basel
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-030-00241-1
DOI
10.1007/978-3-030-00241-1
Hardcover ISBN
978-3-030-00240-4
Edition Number
1
Number of Pages
VI, 191
Topics