# American Institute of Mathematical Sciences

2011, 18: 54-60. doi: 10.3934/era.2011.18.54

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

 1 Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany 2 School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Received  February 2011 Revised  April 2011 Published  July 2011

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.

Citation: Hermann Köenig, Vitali Milman. Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$. Electronic Research Announcements, 2011, 18: 54-60. doi: 10.3934/era.2011.18.54
##### References:
 [1] J. Aczél, "Lectures on Functional Equations and their Applications,", Mathematics in Science and Engineering, 19 (1966).   Google Scholar [2] S. Artstein-Avidan, H. König and V. Milman, The chain rule as a functional equation,, Journ. Funct. Anal., 259 (2010), 2999.  doi: 10.1016/j.jfa.2010.07.002.  Google Scholar [3] H. König and V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule,, preprint., ().   Google Scholar

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##### References:
 [1] J. Aczél, "Lectures on Functional Equations and their Applications,", Mathematics in Science and Engineering, 19 (1966).   Google Scholar [2] S. Artstein-Avidan, H. König and V. Milman, The chain rule as a functional equation,, Journ. Funct. Anal., 259 (2010), 2999.  doi: 10.1016/j.jfa.2010.07.002.  Google Scholar [3] H. König and V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule,, preprint., ().   Google Scholar
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