Approximation of rough functions

Abstract

For given p ∈ [1,∞] and g ∈ Lᵖ(R), we establish the existence and uniqueness of solutions f ∈ Lᵖ(R), to the equation f (x) − a f (bx) = g(x), where a ∈ R, b ∈ R\ {0}, and |a| ̸= |b|1/p. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established.