Abstract
We 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 ofH a(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.
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Lau, KS., Wang, J. Mean quadratic variations and Fourier asymptotics of self-similar measures. Monatshefte für Mathematik 115, 99–132 (1993). https://doi.org/10.1007/BF01311213
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DOI: https://doi.org/10.1007/BF01311213