On the Weierstrass-Mandelbrot fractal function

Abstract

The function W ( t ) ≡ ∑ n = − ∞ ∞ [ ( 1 − e i γ n t ) e i ϕ n ] γ ( 2 − D ) n ( 1 < D < 2 , γ > 1 , ϕ n = arbitrary phases ) is continuous but non-differentiable and possesses no scale. The graph of ReW or Im W has Hausdorff-Besicovitch (fractal) dimension D. Choosing Ø n = 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 Ø n 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 Ø n , for 0 ≤ t ≤ 1 and the magnified range 0.30 ≤ t ≤ 0.31. The Weierstrass spectrum y n can be generated by the energy levels of the quantum-mechanical potential — A / x 2 ,where A = 4π 2 /In 2 y.