Overview
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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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
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.
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Authors and Affiliations
Bibliographic Information
Book Title: Operator Relations Characterizing Derivatives
Authors: Hermann König, Vitali Milman
DOI: https://doi.org/10.1007/978-3-030-00241-1
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Hardcover ISBN: 978-3-030-00240-4Published: 12 October 2018
Softcover ISBN: 978-3-030-13096-1Published: 10 December 2019
eBook ISBN: 978-3-030-00241-1Published: 03 October 2018
Edition Number: 1
Number of Pages: VI, 191
Topics: Difference and Functional Equations, Operator Theory, Real Functions