## Thursday, 18 April 2013

### Functions

Can we Find any two Distinct Functions
$$f:\Re \rightarrow \Re$$ such that

$$f\left ( x \right )=f\left (\frac{x}{2}\right )$$

2. In general $$f(x)=Const$$ is one such function.
3. We can engineer a solution pretty easily here. Consider some $m(x)$ (we'll talk about what this is later).
$$L(x)= m(4x) + m(2x) + m(x) + m(\frac{x}{2}) + m(\frac{x}{8}) + ...$$
It's easy to see that formally speaking $$L(x/2) = L(x)$$. Now that series I gave seems hopeless at first since it consider values $m$ at 0 and infinity. That is fine, let us consider $m(x) = x^3e^{-x^2}$. This rapidly goes to 0 as the $|x|$ gets very large, as well as very small. Then substituting that back into the definition of $L$ gives you a function that is periodic with respect to doubling the period (at least over the reals).
4. The $L(x)$ in my earlier comment should be $L(x) = ... m(4x) ...$ that is, it is infinite in both directions