Local moments and inverse problem of fractal measures

An approach to the inverse problem for fractal measures based on local moment sampling is proposed. It is shown that, given the local moments of a linear Cantor system (LCS) for a sequence of partitions T/sub /r related to the dynamical partitions, one can determine another LCS with q equal scale maps on each interval of T/sub /r having the 2q-2 first local moments. The measure of the LCS so determined converges to the target measure as r goes to infinity while q is kept fixed. The numerical examples show that q=2, for which a simple algorithm is given, is adequate for a reconstruction of the measure within an error which decreases according to a power law with the order of the partition. The extensions to arbitrary fractal measures are possible and only require a clever strategy in the choice of the partitions.

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