© 2006 by Princeton University Press. All Rights Reserved. Using block spin assignments, we construct an L2-orthonormal basis consisting of dyadic scaiings and translates of just a finite number of functions. These functions have exponential localization, and for even values of a construction parameter M one can make them class CM-l with vanishing moments up to order M inclusive. Such a basis has an important application to phase cell cluster expansions in quantum field theory.