Gibbs’ Phenomenon for Nonnegative Compactly Supported Scaling Vectors

This paper considers Gibbs’ phenomenon for scaling vectors in L2(R). We first show that a wide class of multiresolution analyses suffer from Gibbs’ phenomenon. To deal with this problem, in [11], Walter and Shen use an Abel summation technique to construct a positive scaling function Pr, 0 < r < 1, from an orthonormal scaling function φ that generates V0. A reproducing kernel can in turn be constructed using Pr. This kernel is also positive, has unit integral, and approximations utilitizing it display no Gibbs’ phenomenon. These results were extended to scaling vectors and multiwavelets in [9]. In both cases, orthogonality and compact support were lost in the construction process. In this paper we modify the approach given in [9] to construct compactly supported positive scaling vectors. While the mapping into V0 associated with this new positive scaling vector is not a projection, the scaling vector does produce a Riesz basis for V0 and we conclude the paper by illustrating that expansions of functions via positive scaling vectors exhibit no Gibbs’ phenomenon.