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Multiresolution analysis, Haar bases, and self-similar tilings of ${R}^{n}$. (English) Zbl 0742.42012
Summary: Orthonormal bases for ${L}^{2}\left({R}^{n}\right)$ are constructed that have properties that are similar to those enjoyed by the classical Haar basis for ${L}^{2}\left(R\right)$. For example, each basis consists of appropriate dilates and translates of a finite collection of “piecewise constant” functions. The construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self-similar tilings.

##### MSC:
 42C15 General harmonic expansions, frames 28A80 Fractals