A construction of interpolating wavelets on invariant sets

We introduce the concept of a refinable set relative to a family of contrac tive mappings on a metric space, and demonstrate how such sets are useful t o recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analo gous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interp olants. Several concrete examples of refinable sets which can be used for g enerating interpolatory wavelets are included.