Fractal Dimensions and Singularities of the Weierstrass Type Functions

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 de-where X > 1 , 0 < a < 1 , and g is an almost periodic Lipschitz function of order greater than a , it is shown that the ^-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.