In this note we determine the unique solution to the functional equation
f(x + y)(x − y) = (f(x) − f(y)) (x + y). We require no additional assumptions on the function f. Moreover we solve this functional equation if f is only defined on a finite interval.
The interest in this type of functional equation is motivated by the study of symmetrizing
measures for (the generator of) a L´ evy-driven Ornstein–Uhlenbeck process.