FUNCTION NORMS AND FRACTAL DIMENSION

Using functional norms L(alpha)(f), are introduce a two-parameter norm family L((alpha,beta))(f) by performing sections on the definition do main of f. These norms are used on the difference function f(x)-f(y) t o obtain the operators S(t)au((alpha,beta))(f) which measure the irreg ularity of f. The order of growth of S(t)au((alpha,beta)) at 0 determi nes an irregularity index Delta((alpha,beta))(f). In particular, Delta ((infinity,1))(f) is the fractal dimension of the graph of f. We inves tigate the value of Delta((alpha,beta))(f) for the series f(x) = Sigma (n=0)(infinity)2(-nH)g(2(n)x+Phi(n)), where 0 < H < 1, (Phi(n)) is a r eal-number sequence, and g is a continuous periodic function of period 1.