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Convergence of Vector Subdivision Schemes in Sobolev Spaces

✍ Scribed by Di-Rong Chen; Rong-Qing Jia; S.D Riemenschneider

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
206 KB
Volume
12
Category
Article
ISSN
1063-5203

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✦ Synopsis

In this paper we consider functional equations of the form = α∈Z s a(α) (M•α), where = (φ 1 , . . . , φ r ) T is an r × 1 vector of functions on the s-dimensional Euclidean space, a(α), α ∈ Z s , is a finitely supported sequence of r × r complex matrices, and M is an s ×s isotropic integer matrix such that lim n→∞ M -n = 0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function φ j , j = 1, . . . , r, belonging to the Sobolev space W k p (R s )? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Q a associated with the sequence a is defined by

Let 0 be a nontrivial r × 1 vector of compactly supported functions in W k p (R s ). The iteration scheme n = Q a n-1 , n = 1, 2, . . . , is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. These conditions are determined by the matrix A(0) = m -1 α∈Z s a(α), m = | det M|, and are related to polynomial reproducibility and the classical Strang-Fix conditions.

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