FRACTAL DIMENSIONS AND SINGULARITIES OF THE WEIERSTRASS TYPE FUNCTIONS

A new type of fractal measures K(s), 1 less-than-or-equal-to s less-than-or-equal-to 2, defined on the subsets of the graph of a continuous function is introduced. The K-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) = SIGMA0infinity lambda(-alphai) g(lambda(i)x) , where lambda > 1 , 0 < alpha < 1, and g is an almost periodic Lipschitz function of order greater than alpha, it is shown that the K-dimension of the graph of W equals to 2 - alpha, 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.