On an operator preserving lacunarity of Fourier series

Summary.

The operator F, given by $F [\varphi] (x) := \sum\limits_{k = 0}^\infty\,2^{- k} \varphi (2^k x)$, is a continuous automorphism on the Banach space of real bounded functions. The operator equation $F^n [\varphi] = \psi$ is solved explicitly and it is shown that F preserves Hadamard type lacunarity of Fourier series. As a consequence we get for instance that $F^n [c], c (x) = \cos (2 \pi x)$, is continuous but nowhere differentiable for any $n \in \Bbb N$.