Full length articleApproximation of rough functions

For given p∈[1,∞] and g∈Lp(R), we establish the existence and uniqueness of solutions f∈Lp(R), to the equation f(x)−af(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.