Vanishing moments for scaling vectors

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 ϕ i . We then construct an L 2 ( ℠) orthogonal, symmetric scaling vector with one vanishing moment having minimal support.