We consider the solutions of refinement equations written in the form$$\varphi \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi \left( {Mx - \alpha } \right) + g\left( x \right),\,\,\...We consider the solutions of refinement equations written in the form$$\varphi \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi \left( {Mx - \alpha } \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$where the vector of functions } = (}1, ..., }r)T is unknown, g is a given vector of compactly supported functions on A^s, a is a finitely supported sequence of r 2 r matrices called the refinement mask, and M is an s 2 s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask a, g, and dilation M generates a sequence }n, n = 1, 2, ..., by the iterative process$$\varphi _n \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi _{n - 1} \left(Mx - \alpha \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$from a starting vector of function }0. We characterize the Lp-convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.展开更多
In this paper, we shall study the solutions of functional equations of the formΦ=∑α∈Zsa（α）Φ（M.-α）where is an r × 1 column vector of functions on the s-dimensional Euclidean space,a ：= （a（a））α∈Zs...In this paper, we shall study the solutions of functional equations of the formΦ=∑α∈Zsa（α）Φ（M.-α）where is an r × 1 column vector of functions on the s-dimensional Euclidean space,a ：= （a（a））α∈Zs is an exponentially decaying sequence of r × r complex matrices called refinement mask and M is an s × s integer matrix such that limn∞ M-n =0. We axe interested in the question, for a mask a with exponential decay, if there exists a solution ~ to the functional equation with each function φj, j = 1,... ,r, belonging to L2（Rs） and having exponential decay in some sense？ Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by展开更多
The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a...The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.展开更多
It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply s...It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply sufficient conditions for the convergence, which will cover a substantially large class of schemes.展开更多
基金supported by NSF of China under Grant No.10071071
文摘We consider the solutions of refinement equations written in the form$$\varphi \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi \left( {Mx - \alpha } \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$where the vector of functions } = (}1, ..., }r)T is unknown, g is a given vector of compactly supported functions on A^s, a is a finitely supported sequence of r 2 r matrices called the refinement mask, and M is an s 2 s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask a, g, and dilation M generates a sequence }n, n = 1, 2, ..., by the iterative process$$\varphi _n \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi _{n - 1} \left(Mx - \alpha \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$from a starting vector of function }0. We characterize the Lp-convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.
基金Supported by National Natural Science Foundation of China(Grant Nos.11101120, 11171299 and 11001247)Fundamental Research Funds for the Central Universities
文摘In this paper, we shall study the solutions of functional equations of the formΦ=∑α∈Zsa（α）Φ（M.-α）where is an r × 1 column vector of functions on the s-dimensional Euclidean space,a ：= （a（a））α∈Zs is an exponentially decaying sequence of r × r complex matrices called refinement mask and M is an s × s integer matrix such that limn∞ M-n =0. We axe interested in the question, for a mask a with exponential decay, if there exists a solution ~ to the functional equation with each function φj, j = 1,... ,r, belonging to L2（Rs） and having exponential decay in some sense？ Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by
基金The paper is supported financially by State Nature Foundation (No.40004003) Major State Basic Research Program of People's Republic of China (G1999032803).
文摘The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.
基金Supported by Zhejiang Provincial Natural Science Foundation of China (Grant Nos. Y1100440, Y1110491)Science & Technology Program of Zhejiang Province (Grant No. 2009C34006)+1 种基金Foundation of Zhejiang Educational Committee (Grant No. Y201018286)Major Science & Technology Projects of Zhejiang Province (Grant No. 2011C11050)
文摘It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply sufficient conditions for the convergence, which will cover a substantially large class of schemes.
