On the Support Properties of Scaling Vectors

Abstract

C. K. Chui and J. Z. Wang [J. Approx. Theory71(1992), 263–304] derived support properties for a scaling function generating a function space V0⊆ L2(open face R). Motivated by this work, we consider support properties for scaling vectors. T. N. T. Goodman and S. L. Lee [Trans. Amer. Math. Soc.342,No. 1 (Mar. 1994), 307–324] derived necessary and sufficient conditions for the scaling vector {φ1, … , φr}, r ≥1, to form a Riesz basis for V0and develop a general theory for spline wavelets of multiplicity r > 1. We consider conditions under which linear combinations of scaling functions generate V0. These conditions also characterize all other scaling vectors that generate the same V0. In addition, we describe the scaling vectors of minimal support for V0. Next, we give sufficient conditions on the two-scale symbol for scaling vectors under which a given matrix refinement equation can be solved. A spline-wavelet example illustrates these results. For the single scaling function φ, the support of φ is characterized by the degree of the two-scale symbol. The situation is more complicated in the scaling vector case. We prove a result that gives the support of the scaling vector under certain conditions on the coefficient matrices. This result is illustrated by an example of fractal wavelets derived by J. Geronimo, D. Hardin, and P. Massopust [J. Approx. Theory78,No. 3 (1994), 373–401].