On the Weierstrass-Mandelbrot fractal function

Abstract

The function W(t)≡∑n=−∞∞[(1−eiγnt)eiϕn]γ(2−D)n(1<D<2,γ>1,ϕn=arbitraryphases) is continuous but non-differentiable and possesses no scale. The graph of ReW or Im W has Hausdorff-Besicovitch (fractal) dimension D. Choosing Øn = un gives a deterministic W the scaling properties of which can be studied analytically in terms of a representation obtained by using the Poisson summation formula. Choosing Øn random gives a stochastic IF whose increments W( t +r) — W (t) are statistically stationary, with a mean square which, as a function of r, is smooth if 1.0 < D < 1.5 and fractal if 1.5 < D < 2.0. The properties of IF are illustrated by computed graphs for several values of D (including some ‘marginal’ cases = 1 where the series for W converges) and several values of y, with deterministic and random Øn, for 0 ≤ t ≤ 1 and the magnified range 0.30 ≤ t ≤ 0.31. The Weierstrass spectrum yn can be generated by the energy levels of the quantum-mechanical potential — A / x2,where A = 4π2/In2y.