Properties of an Operator Acting on the Space of Bounded Real Functions and Certain Subspaces

Abstract

Let φ: ℝ → ℝ be bounded and α ∈ (0,1), β∈ (0, ∞). Then

$$F\left[ \varphi \right]:\mathbb{R} \to \mathbb{R},F\left[ \varphi \right]\left( x \right): = \sum\limits_{k = 0}^\infty {{\alpha ^k}} \varphi \left( {{\beta ^k}x} \right)$$

defines a two parameter family of operators F = F(α, β) acting on the Banach space of bounded real functions. It turns out that F is a continuous Banach space automorphism.

F and F ^-1 are closely related to a de Rham type functional equation and the eigenspaces of F and F ^-1 as well as their restrictions to certain subspaces are solution spaces of systems of iterative functional equations.