caa a22 4500 46793908X CHVBK 20180406153007.0 cr unu---uuuuu 170328e20060101xx s 000 0 eng 10.1007/s10444-004-7615-2 doi (NATIONALLICENCE)springer-10.1007/s10444-004-7615-2 Han Bin Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada aut Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix [Elektronische Daten] [Bin Han] In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wpk(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 Wpk(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 φ∈Lp(ℝs) (φ∈C(ℝs) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,Lp(ℝs)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143-167) in the univariate setting to the multivariate setting. Springer, 2006 vector refinement equation nationallicence refinable function vector nationallicence Sobolev space nationallicence rate of convergence nationallicence smoothness nationallicence cascade algorithm nationallicence Advances in Computational Mathematics Kluwer Academic Publishers 24/1-4(2006-01-01), 375-403 1019-7168 24:1-4<375 2006 24 10444 https://doi.org/10.1007/s10444-004-7615-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10444-004-7615-2 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Han Bin Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada aut NATIONALLICENCE 773 0- Advances in Computational Mathematics Kluwer Academic Publishers 24/1-4(2006-01-01), 375-403 1019-7168 24:1-4<375 2006 24 10444 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer