Can a function $f: \mathbb{R} \to \mathbb{R}$ be uniquely determined from its symmetric components about two different points? If so how?
Not sure if I'm formulating this correctly, but I've got a problem that looks something like this:
$f(x) = S_a(x)+A_a(x) = S_b(x)+A_b(x)$
where $~~S_a(a+x)=S_a(a-x)$, $~~A_a(a+x) = -A_a(a-x)$,
and $~~~~~~S_b(b+x)=S_b(b-x)$, $~~A_b(b+x) = -A_b(b-x)$,
$S_a$ and $S_b$ are known, and $a\neq b$.
What is $f(x)$?
Edit: $S_a$ and $A_a$ are the symmetric and anti-symmetric components of $f$ around the point $a$ (as defined above).