On an operator preserving lacunarity of Fourier series (Q1349039)

Let \(B\) be the Banach space of real bounded functions defined on the real line \(\mathbb{R}\) and equipped with the supremum norm. It is known that the operator \(F\) defined by \[ F[\varphi](x):= \sum^\infty_{k=0} 2^{-k}\varphi(2^k x),\quad \varphi\in B,\;x\in\mathbb{R}, \] is a continuous automorphism on \(B\). The present author solves explicitly the operator equation \(F^n[\varphi]= \psi\), \(n\in\mathbb{N}\), and proves that \(F\) preserves an Hadamard type lacunarity of Fourier series. As a consequence, one obtains that if \(\varphi(x):= \cos(2\pi x)\), then \(F^n[\varphi]\) is continuous but nowhere differentiable for any \(n\in\mathbb{N}\).