Smooth PONS

PONS(tm) is a basis which satisfies all of the fundamental properties of th e Walsh functions (each element is piecewise constant, rakes on only the va lues +/-1, and can be efficiently computed via a fast transform) plus three additional properties that are false for the Walsh functions: PONS is opti mal with respect to a global uncertainty principle; all PONS elements have uniformly bounded crest factors; and all PONS elements are QMF's. In 1991, Ingrid Daubechies asked whether there exists a smooth basis satisfying the global uncertainty principle property. In this article we show how to trans form any basis into another basis by applying the PONS construction, thereb y providing an affirmative answer to this question.