Volume 25, pp. 138-157, 2006.

New constructions of piecewise-constant wavelets

Youngmi Hur and Amos Ron


The classical Haar wavelet system of $L_2({\bf R}^n)$ is commonly considered to be very local in space. We introduce and study in this paper piecewise-constant framelets (PCF) that include the Haar system as a special case. We show that any bi-framelet pair consisting of PCFs provides the same Besov space characterizations as the Haar system. In particular, it has Jackson-type performance $s_J=1$ and Bernstein-type performance $s_B=0.5$. We then construct two PCF systems that are either, in high spatial dimensions, far more local than Haar, or are as local as Haar while delivering better performance: $s_J=s_B=1$. Both representations are computed and inverted by fast algorithms.

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Key words

frames, framelets, wavelets, Haar wavelets, piecewise-constant wavelets, PCF, Besov spaces, Unitary Extension Principle

AMS subject classifications

42C15, 42C40

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