Fractal Dimensions and Singularities of the Weierstrass Type Functions

Tian-You Hu, Ka-Sing Lau
1993 Transactions of the American Mathematical Society  
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 thê -dimension of the graph of W equals to 2 -a , this conclusion is also equivalent
more » ... certain rate of the local oscillation of the function. Some problems on the ' knot ' points and the nondifferentiability of W are also discussed.
doi:10.2307/2154398 fatcat:fent3icas5epxhukkpstj7f5ru