On the Support Properties of Scaling Vectors

✍ Scribed by Peter R. Massopust; David K. Ruch; Patrick J. Van Fleet

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 210 KB
• Volume: 3
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1006/acha.1996.0018

✦ Synopsis

304] derived support properties for a scaling function generating a function space V 0 ⊆ L 2 (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 {φ 1 , . . . , φ r }, r 1, to form a Riesz basis for V 0 and develop a general theory for spline wavelets of multiplicity r > 1. We consider conditions under which linear combinations of scaling functions generate V 0 . These conditions also characterize all other scaling vectors that generate the same V 0 . In addition, we describe the scaling vectors of minimal support for V 0 . 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.