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The Resource Operator Relations Characterizing Derivatives, by Hermann König, Vitali Milman, (electronic resource)

Operator Relations Characterizing Derivatives, by Hermann König, Vitali Milman, (electronic resource)

Creator
Author
Contributor
Summary
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 C^k-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.--
Language
eng
Work
Publication
Extent
1 online resource (VI, 191 pages)
Contents
  • Introduction
  • Regular Solutions of Some Functional Equations
  • The Leibniz Rule
  • The Chain Rule
  • Stability and Rigidity of the Leibniz and the Chain Rules
  • The Chain Rule Inequality and its Perturbations
  • The Second-Order Leibniz rule
  • Non-localization Results
  • The Second-Order Chain Rule
  • Bibliography
  • Subject Index
  • Author Index
Isbn
9783030002411
Link
https://eui.idm.oclc.org/login?url=https://doi.org/10.1007/978-3-030-00241-1
Instance
Label
Operator Relations Characterizing Derivatives
Title
Operator Relations Characterizing Derivatives
Statement of responsibility
by Hermann König, Vitali Milman
Creator
Contributor
Author
Subject
Language
eng
Summary
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 C^k-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.--
Assigning source
Provided by publisher
http://library.link/vocab/creatorName
König, Hermann
Dewey number
515.625
Image bit depth
0
Literary form
non fiction
Nature of contents
dictionaries
http://library.link/vocab/relatedWorkOrContributorName
Milman, Vitali
Series statement
  • Springer eBooks
  • Springer eBooks.
http://library.link/vocab/subjectName
  • Functional equations
  • Operator theory
  • Mathematics
Label
Operator Relations Characterizing Derivatives, by Hermann König, Vitali Milman, (electronic resource)
Link
https://eui.idm.oclc.org/login?url=https://doi.org/10.1007/978-3-030-00241-1
Instantiates
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
Introduction -- Regular Solutions of Some Functional Equations -- The Leibniz Rule -- The Chain Rule -- Stability and Rigidity of the Leibniz and the Chain Rules -- The Chain Rule Inequality and its Perturbations -- The Second-Order Leibniz rule -- Non-localization Results -- The Second-Order Chain Rule -- Bibliography -- Subject Index -- Author Index
Control code
978-3-030-00241-1
Dimensions
unknown
Extent
1 online resource (VI, 191 pages)
File format
multiple file formats
Form of item
  • online
  • electronic
Governing access note
Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, non-commercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
Isbn
9783030002411
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1056908463
Label
Operator Relations Characterizing Derivatives, by Hermann König, Vitali Milman, (electronic resource)
Link
https://eui.idm.oclc.org/login?url=https://doi.org/10.1007/978-3-030-00241-1
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
Introduction -- Regular Solutions of Some Functional Equations -- The Leibniz Rule -- The Chain Rule -- Stability and Rigidity of the Leibniz and the Chain Rules -- The Chain Rule Inequality and its Perturbations -- The Second-Order Leibniz rule -- Non-localization Results -- The Second-Order Chain Rule -- Bibliography -- Subject Index -- Author Index
Control code
978-3-030-00241-1
Dimensions
unknown
Extent
1 online resource (VI, 191 pages)
File format
multiple file formats
Form of item
  • online
  • electronic
Governing access note
Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, non-commercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
Isbn
9783030002411
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1056908463

Subject

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