Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wk p (Rs )(1 p ) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space Wk p (Rs ) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp(1 p ) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector Lp(Rs ) ( C(Rs ) when p = ) satisfies a refinement equation with a finitely supported matrix mask, then all the components of must belong to a Lipschitz space Lip(, Lp(Rs )) for some > 0. This paper generalizes the results in [R. Q. Jia, K. S. Lau, and D. X. Zhou, J. Fourier Anal. Appl., 7 (2001), pp. 143

• Type: Journal
• Year: 2006
• Where: ADCM
• Authors: Bin Han 0003
• Keywords: ADCM 2006, Function Vector, Refinable Function Vector, Sobolev Space Wk