Approximation of rough functions

For given p ∈ [1, ∞] and g ∈ L p(R), we establish the existence and uniqueness of solutions f ∈ L p(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. c ⃝ 2016 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). MSC: 26A15; 26A18; 26A27; 42A38; 39B12