Ks. Lau et Jr. Wang, MEAN QUADRATIC VARIATIONS AND FOURIER ASYMPTOTICS OF SELF-SIMILAR MEASURES, Monatshefte fuer Mathematik, 115(1-2), 1993, pp. 99-132
We show that the mean quadratic variation of a self-similar measure mu
under certain open set condition exhibits asymptotic periodicity. Thr
ough a generalized Wiener's Tauberian Theorem, we obtain some new iden
tities and equivalences of the mean quadratic variation of a bounded m
easure nu and its Fourier average H(alpha) (T; nu) = = 1/T(n-alpha) in
tegral 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 shar
pen some recent results of Strichartz concerning the asymptotic behavi
or 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.