Characterizations of eigenspaces by functional equations

Summary.

We characterize eigenspaces of restrictions of the operator F given by

$$ F[\varphi ](x) = \sum\limits_{k = 0}^\infty {2^{ - k} \varphi (2^k x)} $$

to some subspaces of the vector space $\{ \varphi :\mathbb{R} \to \mathbb{R};\,F[\varphi ]:\mathbb{R} \to \mathbb{R}{\text{ exists}}\} .$