David K. Ruch

Copyright © 2004 David K. Ruch. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

One advantage of scaling vectors over a single scaling function is the compatibility of symmetry and orthogonality. This paper investigates the relationship between symmetry, vanishing moments, orthogonality, and support length for a scaling vector Φ. Some general results on scaling vectors and vanishing moments are developed, as well as some necessary conditions for the symbol entries of a scaling vector with both symmetry and orthogonality. If orthogonal scaling vector Φ has some kind of symmetry and a given number of vanishing moments, we can characterize the type of symmetry for Φ, give some information about the form of the symbol P(z), and place some bounds on the support of each ϕi. We then construct an L2(ℝ) orthogonal, symmetric scaling vector with one vanishing moment having minimal support.