Convergence of Cascade Algorithms in Sobolev Spaces Associated with Multivariate Refinement Equations

✍ Scribed by Song Li

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 130 KB
• Volume: 257
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.7339

✦ Synopsis

This paper is concerned with multivariate inhomogeneous refinement equations written in the form ϕ x = α∈ s a α ϕ Mx -α + g x x ∈ s , where ϕ is the unknown function defined on the s-dimentional Euclidean space s g is a given compactly supported function on s , a is a finitely supported sequence on s , and M is an s × s dilation matrix with m = det M . Let ϕ 0 be an initial function in the Sobolev space W k 2 s . For n = 1 2 define ϕ n x = α∈ s a α ϕ n-1 Mx -α + g x x ∈ s . In this paper, we give a characterization for the strong convergence in the Sobolev space W k 2 s k ∈ of the cascade sequence ϕ n n∈ for the case in which M is isotropic.