Consider the function $f(x)=\sin( \log x)$ defined over $x>0$.
It has the cool feature that when you plot it, and change the x scale, it's overall shape does not change much. For example if you look at it over the $x$ range $[0,\;0.001]$ or $[0,\;1000]$ it's overall shape doesn't change.
Here is the question: Does there exist any positive real number $c$ that:
$f(c)=f(1)$ and $f(2c)=f(2)$ and $f(3c)=f(3)$ simultaneously?
Can we build a class of functions like $\sin ( \log x)$ that can form a base similar to fourier?