Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z<sub>0</sub> be a subset of Z such that n∈Z<sub>0</sub> implies n+1∈Z<sub>0</sub>....Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z<sub>0</sub> be a subset of Z such that n∈Z<sub>0</sub> implies n+1∈Z<sub>0</sub>. Denote the space of all compactly supported distributions by D’, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G<sub>n</sub> and H<sub>n</sub>, n∈Z<sub>0</sub>, in D’, define the corresponding nonstationary nonhomogeneous refinement equation Φ<sub>n</sub>=H<sub>n</sub>*Φ<sub>n+1</sub>（A.）+G<sub>n</sub> for all n∈Z<sub>0</sub>, （*） where Φ<sub>n</sub>, n∈Z<sub>0</sub>, is in a bounded set of D’. The nonstationary nonhomogeneous refinement equation （*） arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ<sub>n</sub>, n∈Z<sub>0</sub>, of the equation （*）. In fact, we reduce the existence problem to finding a bounded solution F<sub>n</sub> of the linear equations <sub>n</sub>-S<sub>n</sub> <sub>n+1</sub>= <sub>n</sub> for all n∈Z<sub>0</sub>, where the matrices S<sub>n</sub> and the vectors <sub>n</sub>, n∈Z<sub>0</sub>, can be constructed explicitly from H<sub>n</sub> and G<sub>n</sub> respectively. The results above are still new even for stationary nonhomogeneous refinement equations.展开更多
基金supported by the Wavelets Strategic Research ProgramNational University of Singapore+1 种基金 under a grant from the National Science and Technology Board and the Ministry of Education Singapore.
文摘Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z<sub>0</sub> be a subset of Z such that n∈Z<sub>0</sub> implies n+1∈Z<sub>0</sub>. Denote the space of all compactly supported distributions by D’, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G<sub>n</sub> and H<sub>n</sub>, n∈Z<sub>0</sub>, in D’, define the corresponding nonstationary nonhomogeneous refinement equation Φ<sub>n</sub>=H<sub>n</sub>*Φ<sub>n+1</sub>（A.）+G<sub>n</sub> for all n∈Z<sub>0</sub>, （*） where Φ<sub>n</sub>, n∈Z<sub>0</sub>, is in a bounded set of D’. The nonstationary nonhomogeneous refinement equation （*） arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ<sub>n</sub>, n∈Z<sub>0</sub>, of the equation （*）. In fact, we reduce the existence problem to finding a bounded solution F<sub>n</sub> of the linear equations <sub>n</sub>-S<sub>n</sub> <sub>n+1</sub>= <sub>n</sub> for all n∈Z<sub>0</sub>, where the matrices S<sub>n</sub> and the vectors <sub>n</sub>, n∈Z<sub>0</sub>, can be constructed explicitly from H<sub>n</sub> and G<sub>n</sub> respectively. The results above are still new even for stationary nonhomogeneous refinement equations.
