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.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 42C20, 41A25, 39B12.
Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant
G121210654. 相似文献
In this paper we prove a compactness result for compact Kähler Ricci gradient shrinking solitons. If (Mi,gi) is a sequence of Kähler Ricci solitons of real dimension n?4, whose curvatures have uniformly bounded Ln/2 norms, whose Ricci curvatures are uniformly bounded from below and μ(gi,1/2)?A (where μ is Perelman's functional), there is a subsequence (Mi,gi) converging to a compact orbifold (M∞,g∞) with finitely many isolated singularities, where g∞ is a Kähler Ricci soliton metric in an orbifold sense (satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying Kähler Ricci soliton equation in a lifting around singular points). 相似文献
The photolysis reactions of three compounds commonly used as a sunscreen agents, Parsol 1789 (1-[4-(1,1-dimethylethyl)phenyl]-3-(4-methoxyphenyl)-1,3- propanedione), Oxybenzone ((2-hydroxy-4-methoxyphenyl)phenyl-methanone) and Padimate O (2-ethylhexyl-4-(dimethylamino)benzoate), were investigated to provide a chemical background to aid in the understanding of the photosensitization of the sunscreen agents. Photolysis was carried out in cyclohexane for 70–140 h using a mercury vapor lamp (450W) without excluding oxygen.
Irradation of Parsol 1789 in cyclohexane yielded tert-butylbenzene, p-tert-butylbenzoic acid and p-methoxybenzoic acid; products obtained from the combination of the sunscreen with the solvent included the cyclohexyl esters of p-methoxybenzoic acid, p-tert-butylbenzoic acid and methanoic acid; products obtained from the solvent included cyclohexanol, cyclohexanone and dicyclohexyl ether.
Irradiation of Oxybenzone in the cyclohexane for 100 h produced no detectable products by either gas or liquid chromatographic analysis. Oxybenzone was recovered unchanged and no products were observed from the photoinitiated reaction of oxygen with the solvent.
Irradiation of Padimate O in cyclohexane yielded the ethylhexyl esters of p-aminobenzoic acid, p-monomethylaminobenzoic acid and p-dimethylamino (o/m)-methylbenzoic acid, as well as products from the photoinitiated reaction of oxygen with the solvent. 相似文献