## Abstract

The function $W(t) \equiv \sum^\infty_{n=-\infty} \frac{[(1 - e^{i\gamma^nt})e^{i\phi_n}]}{\gamma^{(2-D)n}} (1 < D < 2, \gamma > 1, \phi_n = \text{arbitrary phases}),$ is continuous but non-differentiable and possesses no scale. The graph of ReW or ImW has Hausdorff-Besicovitch (fractal) dimension D. Choosing $\phi_n$ = $\mu n$ 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 $\phi_n$ random gives a stochastic W whose increments W(t+$\tau$)-W(t) are statistically stationary, with a mean square which, as a function of $\tau$, is smooth if 1.0 < D < 1.5 and fractal if 1.5 < D < 2.0. The properties of W are illustrated by computed graphs for several values of D (including some 'marginal' cases D = 1 where the series for W converges) and several values of $\gamma$, with deterministic and random $\phi_n$, for 0 $\leqslant$ t $\leqslant$ 1 and the magnified range 0.30 $\leqslant$ t $\leqslant$ 0.31. The Weierstrass spectrum $\gamma^n$ can be generated by the energy levels of the quantum-mechanical potential -A/x$^2$, where $A = \frac{1}{4} + 4\pi^2/\ln^2\gamma.$