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.
