Stability and Rank Properties of Matrix Subdivision Schemes

Subdivision schemes with matrix masks are a natural extension of the well studied case of subdivision schemes with scalar masks Such schemes arise in the analysis of multivariate scalar schemes in subdivision processes corresponding to shift invariant spaces generated by more than one function in geometric modeling where each component of the curve surface is designed by a di erent linear combination of the control points and in the case of schemes which interpolate function and derivatives values simultaneously The limit of a matrix subdivision scheme can be expressed as a combination of shifts of a re nable matrix function It is shown that if is stable in the sense of a new stability notion for matrix valued functions then the scheme is uniformly convergent Also it is shown that the stability of a maximal submatrix of is related with the linear dependence of its rows and hence of any vector valued function generated by the subdivision scheme Finally it is shown that by proper renormalization of the process relative to the vanishing rows of it is possible to generate vector limit functions with components which are the rst derivative of certain linear combinations of the other components The same approach allows to analyze the smoothness of x Introduction Matrix subdivision schemes play an important role in the analysis of multivari ate subdivision schemes in the construction of multiple knot splines and in Her mite type subdivision They also have strong connections with multiresolution approximation of multiplicity more than and multi wavelets The purpose of this work is to understand the characteristics of the di erent types of matrix subdivision schemes Advanced Topics in Multivariate Approximation F Fontanella K Jetter and P J Laurent eds pp Copyright oc by World Scienti c Publishing Co Inc ISBN xxx All rights of reproduction in any form reserved A Cohen N Dyn and D Levin A uniform stationary matrix subdivision scheme is de ned by a set of real n n matrix coe cients fAj j ZZg with a nite number of non zero Aj s generating control points in IR f ff j j ZZg k recursively by f i X