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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Self-similar measures and their Fourier transforms. II


Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 336 (1993), 335-361
MSC: Primary 42B10
DOI: https://doi.org/10.1090/S0002-9947-1993-1081941-2
MathSciNet review: 1081941
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Abstract: A self-similar measure on $ {{\mathbf{R}}^n}$ was defined by Hutchinson to be a probability measure satisfying $ ({\ast})$

$\displaystyle \mu = \sum\limits_{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}} $

, where $ {S_j}x = {\rho _j}{R_j}x + {b_j}$ is a contractive similarity $ (0 < {\rho _j} < 1,{R_j}$ orthogonal) and the weights $ {a_j}$ satisfy $ 0 < {a_j} < 1,\sum\nolimits_{j = 1}^m {{a_j} = 1} $. By analogy, we define a self-similar distribution by the same identity $ ( {\ast} )$ but allowing the weights $ {a_j}$ to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to $ ( {\ast} )$ among distributions of compact support, and show that the space of such solutions is always finite dimensional.

If $ F$ denotes the Fourier transformation of a self-similar distribution of compact support, let

$\displaystyle H(R) = \frac{1}{{{R^{n - \beta }}}}\int_{\vert x\vert \leq R} {\vert F(x){\vert^2}dx,} $

where $ \beta $ is defined by the equation $ \sum\nolimits_{j = 1}^m {\rho _j^{ - \beta }\vert{a_j}{\vert^2} = 1} $. If $ \rho _j^{{\nu _j}} = \rho $ for some fixed $ \rho $ and $ {\nu _j}$ positive integers we say the $ \{ {\rho _j}\} $ are exponentially commensurable. In this case we prove (under some additional hypotheses) that $ H(R)$ is asymptotic (in a suitable sense) to a bounded function $ \tilde H(R)$ that is bounded away from zero and periodic in the sense that $ \tilde H(\rho R) = \tilde H(R)$ for all $ R > 0$. If the $ \{ {\rho _j}\} $ are exponentially incommensurable then $ {\lim _{R \to \infty }}H(R)$ exists and is nonzero.

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DOI: https://doi.org/10.1090/S0002-9947-1993-1081941-2
Article copyright: © Copyright 1993 American Mathematical Society