Fractal Dimensions and Singularities of the Weierstrass Type Functions

Abstract

Abstract. A new type of fractal measures Xs, 1 < s < 2, defined on the subsets of the graph of a continuous function is introduced. The ^-dimension defined by this measure is 'closer' to the Hausdorff dimension than the other fractal dimensions in recent literatures. For the Weierstrass type functions defined by W(x) = £S°'k-aig()Jx), where X > 1 , 0 < a < 1 , and g is an almost periodic Lipschitz function of order greater than a , it is shown that thê -dimension of the graph of W equals to 2 -a , this conclusion is also equivalent to certain rate of the local oscillation of the function. Some problems on the ' knot ' points and the nondifferentiability of W are also discussed.

Hausdorff Dimension of a Class of Weierstrass Functions

“…From e(1) = 2, there exist 0 ≤ k < < 5 such that (k, ) ∈ E(1, x * ). Now we will show that x * ∈ (3/20, 7/20) and (k, ) = (3,4).…”

“…Recently, Shen [7] proved that the graph of the classical Weierstrass function ∑ ∞ n=0 λ n cos(2πb n x) has Hausdorff dimension 2 + log λ/ log b, for every integer b ≥ 2 and every λ ∈ (1/b, 1), which solved a longstanding conjecture. Some relevant results can be found in [1,[3][4][5]8]. Naturally, we want to study the Hausdorff dimension of the graph of Weierstrass functions with the following form:…”

On a class of Besicovitch functions to have exact box dimension: A necessary and sufficient condition

“…The fractal dimensions of Weierstrass type functions have been investigated extensively (see [1,2,5]). It is well-known that their box dimension equals to s while λ is sufficiently large.…”

Weierstrass functions in Zygmund’s class

“…If α < 1 and g is Lipschitz, it is easy to prove that f is of class C α (see for example [4]). Moreover, studying the oscillations of f , numerous works give the conclusion that in good cases the function f is nowhere differentiable (see for example [1,2,6,7,8,11]). In particular, it is proved in [2] and [1] that, as soon as f is not Lipschitz, there exists a constant C > 0 such that for every interval I of length |I| ≤ 1, osc (f, I) = sup I (f ) − inf I (f ) ≥ C |I| α .…”