The chain rule as a functional equation

摘要

We consider operators T from C~1(R{double-struck}) to C(R{double-struck}) satisfying the "chain rule". T(f o g)=(Tf) o g ·Tg,f,gεC~1(R{double-struck}), and study under which conditions this functional equation admits only the derivative or its powers as solutions. We also consider T operating on other domains like C~k(R{double-struck}) for kεN{double-struck}_0 or k=∞ and study the more general equation T(f o g)=(Tf) o g ·Ag, f,gεC~1(R{double-struck}) where both T and A map C~1(R{double-struck}) to C(R{double-struck}).