A functional equation characterizing the second derivative

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. © 2011 Elsevier Inc. All rights reserved.