On the Construction of Positive Scaling Vectors

Let be a compactly supported, orthonormal scaling function that generates the linear space V0 2 L2(IR). We are motivated by the work of Walter and Shen [7]. In this paper, the authors use an Abel summation technique to build a positive scaling function Pr, 0 < r < 1, for the space 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. A natural question and thus the purpose of this paper is to ask whether the work in [7] can be generalized to the scaling vector setting; and, if so, what advantages are there to be gained. In this paper, we show that such a generalization does indeed exist and an anologous kernel can be constructed. The construction requires that the sum of the integer translates of each component of the scaling vector be nonnegative a requirement not necessarily satis ed by a general scaling vector. We obtain our results by assuming this property is satis ed and then conclude the paper by exhibiting an invertible linear transformation that takes a scaling vector satisfying modest conditions and maps it to one suitable for our construction.