A functional equation characterizing the second derivative

作者：Koenig Hermann; Milman Vitali

来源：Journal of Functional Analysis, 2011, 261(4): 876-896.

DOI：10.1016/j.jfa.2011.02.024

• Chain rule
• Second derivative
• Localization

Consider an operator T : C(2)(R) -> C(R) and isotropic maps A(1), A(2) : C(1)(R) -> C(R) such that the functional equation
T(f circle g) = (Tf) circle g . A(1)g + (A(2)f) o circle g . Tg; f,g is an element of C(2)(R)
is satisfied on C(2)(R). The equation models the chain rule for the second derivative, in which case A(1)g = g'(2) and A(2)f = f'. We show under mild non-degeneracy conditions - which imply that A(1) and A(2) are very different from T - that A(1) and A(2) must be of the very restricted form A(1)f = f' . A(2)f, A(2)f = vertical bar f'vertical bar(p) or sgn(f')vertical bar f'vertical bar(p), with p >= 1, and that any solution operator T has the form
Tf(x)= c A(2)(f(x))/f'(x) f ''(x)+(H(f(x))f'(x) - H(x))A(2)(f(x)), x is an element of R
for some constant c is an element of 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 second 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.

• 出版日期2011-8-15