Regularity of multiwavelets
 C.A. Micchelli,
 Thomas Sauer
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The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for Bsplines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets.
The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary L _{ p } spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on biinfinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a biinfinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable biinfinite vector fields.
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 Title
 Regularity of multiwavelets
 Journal

Advances in Computational Mathematics
Volume 7, Issue 4 , pp 455545
 Cover Date
 19970801
 DOI
 10.1023/A:1018971524949
 Print ISSN
 10197168
 Online ISSN
 15729044
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 subdivision
 refinable functions
 regularity
 multiwavelets
 39B12
 41A15
 41A25
 65D99
 Authors

 C.A. Micchelli ^{(1)}
 Thomas Sauer ^{(2)}
 Author Affiliations

 1. T.J. Watson Research Center, IBM Department of Mathematical Sciences, P.O. Box 218, Yorktown Heights, NY, 10598, USA
 2. Institute of Mathematics, University ErlangenNuremberg, D90537, Erlangen, Germany