I am looking for an operator $U$, that can do this to a function: $Uf(x)=f(2x).$
In particular I am happy if there is an $U$ for the general case: $Uf(x)=f(kx)$.
Does such an operator exist for all $f(x)$? What is the operator for known classes of $f(x)$?
I have so far 1 condition on $f(x)$: $\int_{-\infty}^{\infty} f(x)^2\,dx\text{ converges.}$
I dont really know where to look for such an operator, or what keyphrases here are.