Abstract :
Consider an operator T : C2(R) → C(R) and isotropic maps A1,A2 : C1(R) → C(R) such that the
functional equation
T (f ◦ g) = (Tf ) ◦ g · A1g +(A2f ) ◦ g · Tg; f, g ∈ C2(R)
is satisfied on C2(R). The equation models the chain rule for the second derivative, in which case A1g = g
2
and A2f = f
. We show under mild non-degeneracy conditions – which imply that A1 and A2 are very
different from T – that A1 and A2 must be of the very restricted form A1f = f
· A2f , A2f = |f
|p or
sgn(f
)|f
|p, with p 1, and that any solution operator T has the form
Tf (x) = c
A2(f (x))
f
(x)
f
(x)+
H
f (x)
f
(x)− H(x)
A2
f (x)
, x∈ R
for some constant c ∈ R and some continuous function H. Conversely, any such map T satisfies the functional
equation. Under some natural normalization condition, the only solution of the functional equation is
Tf = f
which means that the composition rule with some normalization condition characterizes the sec-ond derivative. If c = 0, T does not depend on the second derivative. In this case, there are further solutions
of the functional equation which we determine, too.
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