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This article is cited in 6 scientific papers (total in 6 papers)
Rigidity and stability of the Leibniz and the chain rule
Hermann Königa, Vitali Milmanb a Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Kiel, Germany
b School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
Abstract:
We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators $V,T_1,T_2,A\colon C^k(\mathbb R)\to C(\mathbb R)$ satisfy equations of the generalized Leibniz and chain rule type for $f,g\in C^k(\mathbb R)$, namely, $V(f\cdot g)=(T_1f)\cdot g+f\cdot(T_2g)$ for $k=1$, $V(f\cdot g)=(T_1f)\cdot g+f\cdot(T_2g)+(Af)\cdot(Ag)$ for $k=2$, and $V(f\circ g)=(T_1f)\circ g\cdot(T_2g)$ for $k=1$. Moreover, for multiplicative maps $A$, we consider a more general version of the first equation, $V(f\cdot g)=(T_1f)\cdot(Ag)+(Af)\cdot(T_2g)$ for $k=1$. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators $V$, $T_1$ and $T_2$ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, $T(f\circ g)=Tf\circ g\cdot Tg+B(f\circ g,g)$ and $T(f\cdot g)=Tf\cdot g+f\cdot Tg+B(f,g)$, and show under suitable conditions on $B$ in the first case that $B=0$ and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation.
Received in January 2012
Citation:
Hermann König, Vitali Milman, “Rigidity and stability of the Leibniz and the chain rule”, Orthogonal series, approximation theory, and related problems, Collected papers. Dedicated to Academician Boris Sergeevich Kashin on the occasion of his 60th birthday, Trudy Mat. Inst. Steklova, 280, MAIK Nauka/Interperiodica, Moscow, 2013, 198–214; Proc. Steklov Inst. Math., 280 (2013), 191–207
Linking options:
https://www.mathnet.ru/eng/tm3455https://doi.org/10.1134/S0371968513010135 https://www.mathnet.ru/eng/tm/v280/p198
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