Function norms and fractal dimension

Using functional norms $L^\alpha(f)$, we introduce a two-parameter norm family $\La^{(\alpha,\beta)}(f)$ by performing sections on the definition domain of f. These norms are used on the difference function $f(x)-f(y)$ to obtain the operators $\St_\tau^{(\alpha,\beta)}(f)$ which measure the irregularity of f. The order of growth of$\St_\tau^{(\alpha,\beta)}(f)$ at 0 determines an irregularity index $\Delta^{(\alpha,\beta)}(f)$. In particular, $\Delta^{(\infty,1)}(f)$ is the fractal dimension of the graph of f. We investigate the value of $\Delta^{(\alpha,\beta)}(f)$ for the series $f(x)=\sum_{n=0}^\infty 2^{-nH}\,g(2^n\,x+\phi_n)$, where $0 < H < 1$, $(\phi_n)$ is a real-number sequence, and g is a continuous periodic function of period 1.