Find all the possible continuous functions $f : \mathbb{R} \to \mathbb{C}$ such that $|f|=1$ and $$\forall x,y \in \mathbb{R},f(x+y)f(x-y)=(f(x))^2.$$
By taking suitable values for $x,y$ we have $f(0)=1,f(2x)=(f(x))^2,$ also $x \to e^{icx},c \in \mathbb{R}$ verifies the above functional equation. Tried to find $f(nx)$ in term of $f(x),$ but unfortunately didn't find anything.
Any suggestions?
Best Answer
$a=x+y$
$b=x-y$
Because $f(2x)=(f(x))^2$ it is easily seen $f(a)f(b)=f(a+b)$
$f(a)=e^{ig(a)}$
This implies $g(a)+g(b)+k2\pi=g(a+b)$ where $k$ is an integer
define $h(a)=g(a)+k2\pi$
$h(a)+h(b)=g(a)+g(b)+k4\pi$
$h(a)+h(b)=g(a+b)+k2\pi=h(a+b)$
we have $h(a)+h(b)=h(a+b)$ which is Cauchy functional equation with solution $h(a)=ca$ provided $h(a)$ is measurable ($f(a)$ is measurable)
Therefore $f(x)=e^{icx-ik2\pi}=e^{icx}$