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Solving the inverse problem for measures using iterated function systems: a new approach

Part of: Approximations and expansions Classical measure theory General theory of differentiable manifolds

Published online by Cambridge University Press:  01 July 2016

B. Forte  and
E. R. Vrscay
B. Forte*
Affiliation:
University of Waterloo
E. R. Vrscay*
Affiliation:
University of Waterloo
*
* Present address: Facoltà di Scienze MM. FF. e NN. a Càvignal, Università degli Studi di Verona, Strada Le Grazie, 37134 Verona, Italy.
** Postal address: Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.
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Abstract

We present a systematic method of approximating, to an arbitrary accuracy, a probability measure µ on x = [0,1]q , q 1, with invariant measures for iterated function systems by matching its moments. There are two novel features in our treatment. 1. An infinite set of fixed affine contraction maps on , w 2, · ·· }, subject to an ‘ϵ-contractivity' condition, is employed. Thus, only an optimization over the associated probabilities pi is required. 2. We prove a collage theorem for moments which reduces the moment matching problem to that of minimizing the collage distance between moment vectors. The minimization procedure is a standard quadratic programming problem in the pi which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0, 1] are presented.

Keywords

COLLAGE DISTANCEQUADRATIC PROGRAMMINGDATA COMPRESSION

MSC classification

Primary: 28A80: Fractals
Secondary: 41A20: Approximation by rational functions 58A40: Differential spaces

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 800 - 820
Copyright
Copyright © Applied Probability Trust 1995 

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