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 [Contemp. Math. 216 (1998) 63–79], 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 utilizing it display no Gibbs’ phenomenon. These results were extended to scaling vectors and multiwavelets in [Proceedings ofWavelet Analysis and MultiresolutionMethods, 2000, pp. 317–339]. In both cases, orthogonality and compact support were lost in the construction process. In this paper we modify the approach given in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317–339] 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.  2004 Elsevier Inc. All rights reserved.