Mean quadratic variations and Fourier asymptotics of self-similar measures

AbstractWe show that the mean quadratic variation of a self-similar measure μ under certain open set condition exhibits asymptotic periodicity. Through a generalized Wiener's Tauberian Theorem, we obtain some new identities and equivalences of the mean quadratic variation of a bounded measurev and its Fourier average $$H_\alpha (T;v) = \frac{1}{{T^{n - _\alpha } }}\int_{|x| \leqslant T} {|\hat v(x)|^2 dx} (0 \leqslant \alpha \leqslant n)$$ . They are used to sharpen some recent results of Strichartz concerning the asymptotic behavior ofHa(T); μ) asT→∞, where μ is the self-similar measure as above. In the development some results concerning the open set condition are also obtained.