On a Banach space automorphism and its connections to functional equations and continuous nowhere differentiable functions

Abstract

Denote by $ℋ$ the Banach space of functions $φ : ℝ → ℝ$ which are continuous, 1-periodic and even. It turns out that $F : ℋ → ℋ$, given by \[F[φ](x) := \sum_{k=0}^∞ 1/2^kφ(2^kx),\] is a Banach space automorphism. Important properties of $F$ are closely related to a de Rham type functional equation for $F[φ]$, Many continuous nowhere differentiable functions are of the form $F[φ]$, A large part of them can be identified by simple properties of the generating function $φ$.