Fredholm properties of multivariate refinement equations

✍ Scribed by Victor D. Didenko

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 96 KB
• Volume: 30
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.859

✦ Synopsis

Abstract

The solvability of multivariate refinement equations in the Hilbert space L~2~(ℝ^s^) is studied. It is shown that the set of all L~2~‐solutions of such equations either consists of the trivial solution only or it contains a subspace isomorphic to a space L~∞~(S~M~) where S~M~ is a subset of ℝ^s^ with a positive Lebesgue measure. Therefore, the corresponding multivariate refinement operator is Fredholm if and only if it is invertible. Copyright © 2007 John Wiley & Sons, Ltd.