Let $f(t)$ be a characteristic function of some random variable $X$ we know that $\displaystyle f(t) = e^{it-4t^2} \cdot f(\frac{t}{2})$ question is to find all possible functions $f$.
I tried something taking log and got something like $g(t)-g(\frac{t}{2}) = it - 4t^2$ where $g(t) = \ln(f(t))$ and hypothesis is $g(t)$ is quadratic function.
Well, apparently, then $$f(t) = e^{it-4t^2}\cdot e^{i{t\over2}-4\left({t\over2}\right)^2}\cdot f\left(\frac{t}4\right)$$ By expanding $f\left(\frac t4\right)$, then $f\left(\frac t8\right)$ and so on, we get a chain of exponents which simplifies nicely, and some $f\left(\frac{t}{2^n}\right)$. Now, $f(x)$ being a characteristic function tells us something about its behavior at $x\to0$, doesn't it?