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Let a data set {(x~, y~) ~ I x R; i = 0, 1,..., N} be given, where I = [Xo, XN]C R. We introduce iterated function systems whose attractors G are graphs of continuous functions f: I~ R, which interpolate the data according to f(x,)=Yl for i~ {0, 1 ..... N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.

Iterated function systems (i. f. ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i. f. ss and occur as the supports of probability measures associated with functional equations. The existence of certain ‘ p -balanced’ measures for i. f. ss is established, and these measures are uniquely characterized for hyperbolic i. f. ss. The Hausdorff—Besicovitch dimension for some attrac­tors of hyperbolic i. f. ss is estimated with the aid of p -balanced measures. What appears to be the broadest framework for the exactly computable moment theory of p -balanced measures — that of linear i. f. ss and of probabilistic mixtures of iterated Riemann surfaces — is presented. This extensively generalizes earlier work on orthogonal polynomials on Julia sets. An example is given of fractal reconstruction with the use of linear i. f. ss and moment theory.

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The problem of providing succinct approximate descriptions of given bounded subsets of RX can be solved by application of the contraction mapping principle.We present an application of the contraction mapping principle which yields succinct descriptions, approximations, and reconstructions for complicated sets such as fractals (1)

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