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Operator Relations Characterizing Derivatives

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Operator Relations Characterizing Derivatives / by Hermann König, Vitali Milman.
作者:	König, Hermann.
其他作者:	Milman, Vitali.
面頁冊數:	VI, 191 p.online resource. :
Contained By:	Springer Nature eBook
標題:	Difference equations. -
電子資源:	https://doi.org/10.1007/978-3-030-00241-1
ISBN:	9783030002411

Operator Relations Characterizing Derivatives
König, Hermann.

Operator Relations Characterizing Derivatives [electronic resource] /by Hermann König, Vitali Milman. - 1st ed. 2018. - VI, 191 p.online resource.

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.

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.

ISBN: 9783030002411

Standard No.: 10.1007/978-3-030-00241-1doiSubjects--Topical Terms:

527665
Difference equations.

LC Class. No.: QA431

Dewey Class. No.: 515.625

Operator Relations Characterizing Derivatives
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505 0 $a 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.
520 $a 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.
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