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New constructions of piecewise-constant wavelets, Electronic Trans. Numerical Analysis 25

By Youngmi Hur and Amos Ron


Abstract. The classical Haar wavelet system of L2(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 sJ = 1 and Bernstein-type performance sB = 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: sJ = sB = 1. Both representations are computed and inverted by fast algorithms

Topics: Key words. frames, framelets, wavelets, Haar wavelets, piecewise-constant wavelets, PCF, Besov spaces, Unitary Extension Principle
Year: 2006
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