LEADER 03964nam 22005415 450 001 9910300137203321 005 20220413215539.0 010 $a3-030-00241-1 024 7 $a10.1007/978-3-030-00241-1 035 $a(CKB)4100000006999432 035 $a(MiAaPQ)EBC5535801 035 $a(DE-He213)978-3-030-00241-1 035 $a(PPN)231464916 035 $a(EXLCZ)994100000006999432 100 $a20181003d2018 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aOperator relations characterizing derivatives /$fby Hermann König, Vitali Milman 205 $a1st ed. 2018. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2018. 215 $a1 online resource (193 pages) 311 $a3-030-00240-3 327 $aIntroduction -- 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. 330 $aThis 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. 606 $aDifference equations 606 $aFunctional equations 606 $aOperator theory 606 $aFunctions of real variables 606 $aDifference and Functional Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12031 606 $aOperator Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M12139 606 $aReal Functions$3https://scigraph.springernature.com/ontologies/product-market-codes/M12171 615 0$aDifference equations. 615 0$aFunctional equations. 615 0$aOperator theory. 615 0$aFunctions of real variables. 615 14$aDifference and Functional Equations. 615 24$aOperator Theory. 615 24$aReal Functions. 676 $a515.724 700 $aKönig$b Hermann$4aut$4http://id.loc.gov/vocabulary/relators/aut$0535222 702 $aMilman$b Vitali$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910300137203321 996 $aOperator Relations Characterizing Derivatives$91563701 997 $aUNINA