The purpose of this paper is to investigate the mean size formula of wavelet packets （wavelet subdivision tree） on Heisenberg group. The formula is given in terms of the p-norm joint spectral radius. The vector refi...The purpose of this paper is to investigate the mean size formula of wavelet packets （wavelet subdivision tree） on Heisenberg group. The formula is given in terms of the p-norm joint spectral radius. The vector refinement equations on Heisenberg group and the subdivision tree on the Heisenberg group are discussed. The mean size formula of wavelet packets can be used to describe the asymptotic behavior of norm of the subdivision tree.展开更多
In this paper it is proved that L p solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in L p without any assumption on the ...In this paper it is proved that L p solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in L p without any assumption on the stability of the solutions of the refinement equation.A characterization for convergence of subdivision scheme is also given in terms of the refinement mask.Thus a complete answer to the relation between the existence of L p solutions of the refinement equation and the convergence of the corresponding subdivision schemes is given.展开更多
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
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the National Natural Science Foundation of China (10471123 10771190)
文摘The purpose of this paper is to investigate the mean size formula of wavelet packets （wavelet subdivision tree） on Heisenberg group. The formula is given in terms of the p-norm joint spectral radius. The vector refinement equations on Heisenberg group and the subdivision tree on the Heisenberg group are discussed. The mean size formula of wavelet packets can be used to describe the asymptotic behavior of norm of the subdivision tree.
基金Supported by the National Natural Science Foundation of China(1 0 0 71 0 71 )
文摘In this paper it is proved that L p solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in L p without any assumption on the stability of the solutions of the refinement equation.A characterization for convergence of subdivision scheme is also given in terms of the refinement mask.Thus a complete answer to the relation between the existence of L p solutions of the refinement equation and the convergence of the corresponding subdivision schemes is given.
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 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.
