MEAN QUADRATIC VARIATIONS AND FOURIER ASYMPTOTICS OF SELF-SIMILAR MEASURES

We show that the mean quadratic variation of a self-similar measure mu 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 measure nu and its Fourier average H(alpha) (T; nu) = = 1/T(n-alpha) integral absolute-value-of x less-than-or-equal-to T \nu(x)\2dx (0 less-than-or-equal-to alpha less-than-or-equal-to n). They are used to sharpen some recent results of Strichartz concerning the asymptotic behavior of H(alpha) (T; mu) as T --> infinity, where mu is the self-similar measure as above. In the development some results concerning the open set condition are also obtained.