International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 36, Pages 1897-1908

Vanishing moments for scaling vectors

David K. Ruch

Department of Mathematical and Computer Sciences, Metropolitan State College of Denver, Denver 80217, CO, USA

Received 20 August 2003

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.


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() orthogonal, symmetric scaling vector with one vanishing moment having minimal support.