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

Abstract. The classical Haar wavelet system of 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 and Bernstein-type performance ff fi. 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: flffi ! . Both representations are computed and inverted by fast algorithms