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Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$

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  • Let $T:C^1(RR)\to C(RR)$ be an operator satisfying the derivation equation

    $T(f\cdot g)=(Tf)\cdot g + f \cdot (Tg),$

    where $f,g\in C^1(RR)$, and some weak additional assumption. Then $T$ must be of the form

    $(Tf)(x) = c(x) \, f'(x) + d(x) \, f(x) \, \ln |f(x)|$

    for $f \in C^1(RR), x \in RR$, where $c, d \in C(RR)$ are suitable continuous functions, with the convention $0 \ln 0 = 0$. If the domain of $T$ is assumed to be $C(RR)$, then $c=0$ and $T$ is essentially given by the entropy function $f \ln |f|$. We can also determine the solutions of the generalized derivation equation

    $T(f\cdot g)=(Tf)\cdot (A_1g) + (A_2f) \cdot (Tg), $

    where $f,g\in C^1(RR)$, for operators $T:C^1(RR)\to C(RR)$ and $A_1, A_2:C(RR)\to C(RR)$ fulfilling some weak additional properties.

    Mathematics Subject Classification: Primary: 26A24; Secondary: 26A06.


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  • [1]

    J. Aczél, "Lectures on Functional Equations and their Applications," Mathematics in Science and Engineering, 19, Academic Press, New York-London, 1966.


    S. Artstein-Avidan, H. König and V. Milman, The chain rule as a functional equation, Journ. Funct. Anal., 259 (2010), 2999-3024.doi: 10.1016/j.jfa.2010.07.002.


    H. König and V. MilmanCharacterizing the derivative and the entropy function by the Leibniz rule, preprint.

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