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, Academic Press, New York-London, 1966.

[2]

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.

[3]

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

show all references

References:
[1]

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

[2]

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.

[3]

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

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