DERIVATIVE AND ENTROPY: THE ONLY DERIVATIONS FROM C-1(R) TO C(R)

作者：Koeenig Hermann; Milman Vitali

来源：Electronic Research Announcements in Mathematical Sciences, 2011, 18: 54-60.

DOI：10.3934/era.2011.18.54

• Derivation
• Entropy function
• Functional Equations

Let T : C-1( R) -> C( R) be an operator satisfying the derivation equation
T(f . g) = (Tf). g + F. (Tg),
where f, g is an element of C-1(R), and some weak additional assumption. Then T must be of the form
(T f)(x) = c(x) f(1)(x) + d(x) f(x) ln vertical bar f(x)vertical bar
for f is an element of C-1(R), x is an element of R, where c, d is an element of C(R) are suitable continuous functions, with the convention 0 ln 0 = 0. If the domain of T is assumed to be C(R), then c = 0 and T is essentially given by the entropy function f in |f|. We can also determine the solutions of the generalized derivation equation
T(f . g) = (Tf). A(1g)) + (A(2)f) . (Tg),
where f, g is an element of C-1(R), for operators T : C-1(R) -> C(R) and A1, A2 : C(R) -> C(R) fulfilling some weak additional properties.

• 出版日期2011