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
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