Weierstrass function

Various means of constructing self-similar surfaces are known.33 Some of them do not allow one to produce different realizations of surface profiles, for example, by making use of the Weierstrass function. These methods should be avoided in the current context because it would be difficult to make statistically meaningful statements without averaging over a set of statistically independent simulations. An appropriate method through which to construct self-similar surfaces i s to use a representation of the height profile h(x) via its Fourier transform h(q). 

In their theoretical work,43 the various self-affine fractal interfaces were mathematically constructed employing the Weierstrass function /ws(x), 151>152... 

Fractal Dimensions of the Profiles h(x) at Various Morphological Amplitudes rj in h(x) = 77/wsCv) Determined by the Current Transient Technique (2nd Column) and the Triangulation Method (3rd Column). Here, /ws(-v) Means the Weierstrass Function with a Self-Affine Fractal Dimension t/Fsa = 1.5 ... 

In terms of Weierstrass functions, V and , these latter quantities are... 

The function cri appearing in Eq. (83) is an even quasi-periodic function (while cr, another Weierstrass function, is odd) defined by the relation [14]... 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

Richardson, in his 1926 investigation of turbulence, observed that the velocity field of the atmospheric wind is so erratic that it probably cannot be described by an analytic function [79]. He suggested a Weierstrass function as a candidate to represent the velocity field, since the function is continuous everywhere, but nowhere differentiable, properties he observed in the wind-field data. Here we investigate a generalization of the Weierstrass function in order to simplify some of the discussion ... 

Another example of a nowhere differentiable function is the Weierstrass function, defined by [19]... 

Figure 6.3 Weierstrass functions (6.23) with different degrees of roughness.

B.R. Hunt. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Am. Math. Soc., 126 791, 1998. 

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