New constructions of piecewise-constant wavelets

The classical Haar wavelet system of L₂(ℝⁿ) 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.