Using the Matrix Refinement Equation for the Construction of Wavelets II: Smooth Wavelets on [0,1]

Abstract

This paper continues the work in [4] on constructing orthogonal bases on the interval [0,1] by using the matrix refinement equation and the two basic operations of translation and scale. We call the elements of these bases wavelets. Here we amplify on the applicability of our method and construct smooth wavelets with and without boundary conditions. That is, we describe a procedure to recursively generate orthonormal bases with any prescribed number of continuous derivatives. As a caveat to the reader we reiterate our remark above that this paper is a continuation of our work in [4] and therefore some familiarity with [4] is assumed.