Solution for this equation

Question

I have been working on a random function I found with some data I simulated and was wondering if anyone has an idea how to, or perhaps can even find a solution for it. The relationship is: $$ f(x) = 2\cdot f(4x) $$ I have no idea if it's even possible to find an explicit solution for f(x) from this, on the other hand, maybe it's really easy, but I am completely stumped. Anyone want to have a go?

Answer 1

One possibility that comes straight to my mind is $f(x)=|x|^{-1/2}$ (for $x\neq0$). Then $$ f(4x)=\frac{1}{|4x|^{1/2}}=\frac{1}{2}f(x) $$ as you desire.

More generally, you can look for homogeneous functions of degree $-1/2$, i.e. functions $h(x)$ that satisfy $$ h(\lambda x)=\lambda^{-1/2}h(x) $$ for any $\lambda>0$ and then take $\lambda=4$.

Hope it helps.

Answer 2

If $f \in \mathcal{C}([0,\alpha[)$ where $\alpha>0$.

We have :

$$\forall x \in [0,\alpha[, \forall n \in \mathbb{N}, f(x) = f(x/4)/2 = f(x/16)/4 =... = f(x/4^n)/2^n $$ then by continuity $\forall x \in [0,\alpha[, f(x) = 0$.

Well it might not answer your expectations though.