This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（...This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained.展开更多
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
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
The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 ...The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius.展开更多
An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampli...An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation , a piece-wise linear function , and posseses an explicit computation formula . Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function .展开更多
In this paper, some characterizations on the convergence rate of both the homogeneous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.
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
Based on complex variables and conformal mapping, the elastic wave scat- tering and dynamic stress concentrations in the plates with two holes are studied by the refined dynamic equation of plate bending. The problem ...Based on complex variables and conformal mapping, the elastic wave scat- tering and dynamic stress concentrations in the plates with two holes are studied by the refined dynamic equation of plate bending. The problem to be solved is changed to a set of infinite algebraic equations by an orthogonM function expansion method. As examples, under free boundary conditions, the numerical results of the dynamic moment concen- tration factors in the plates with two circular holes are computed. The results indicate that the parameters such as the incident wave number, the thickness of plates, and the spacing between holes have great effects on the dynamic stress distributions. The results are accurate because the refined equation is derived without any engineering hypothese.展开更多
In this paper, based on complex variables and conformal mapping methods, using the refined dynamic equation of plates, elastic wave scattering and dynamic stress concentrations in plates with two cutouts were studied....In this paper, based on complex variables and conformal mapping methods, using the refined dynamic equation of plates, elastic wave scattering and dynamic stress concentrations in plates with two cutouts were studied. Applying the orthogonal function expansion method, the problem to be solved can be reduced into the solution of a set of infinite algebraic equations. According to free boundary conditions, numerical results of dynamic moment concentration factors in thick plates with two circular cutouts analyze that: there will be more complex interaction changes between two-cutout situation than single cutout situation. In the case of low frequency or high frequency and thin plate, the hole-spacing in the absence of coupling interactions was larger or smaller. The numerical results and method can be used to analyze the dynamics and strength of plate-like structures.展开更多
This paper is concerned with seeking the general solutions of matrix equation M（ξ）M＊ （ξ） = Is for the construction of multiple channel biorthogonal wavelets, provided that some special solution of its is known.
Based on the refined dynamic equation of stretching plates, the elastic tensio compression wave scattering and dynamic stress concentrations in the thick plate with two cutouts are studied. In view of the problem that...Based on the refined dynamic equation of stretching plates, the elastic tensio compression wave scattering and dynamic stress concentrations in the thick plate with two cutouts are studied. In view of the problem that the shear stress is automatically satisfied under the free boundary condition, the generalized stress of the first-order vanishing moment of shear stress is considered. The numerical results indicate that, as the cutout is thick, the maximum value of the dynamic stress factor obtained using the refined dynamic theory is 19% higher than that from the solution of plane stress problems of elastic dynamics.展开更多
基金Supported by the National Natural Science Foundation of China (10071071)
文摘This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained.
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.
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.
文摘The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius.
基金the NSF of Henan Province (984051900)the NSF of Henan Education Committee (98110015)the Excellent Teacher Foundation of High School in Henan Province
文摘An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation , a piece-wise linear function , and posseses an explicit computation formula . Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function .
基金This project is partially supported by Zhejiang Provincial Natural Science Foundation of Chinathe second author is also supported by Postdoctral Fellowship Foundation of China in partThis paper is based on the report "Studies on Wavelet Analysis in Z
文摘A review of the advance in the theory of wavelet analysis in recent years is given.
文摘In this paper, some characterizations on the convergence rate of both the homogeneous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.
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Project supported by the National Natural Science Foundation of China(Nos.51378451 and 51378245)
文摘Based on complex variables and conformal mapping, the elastic wave scat- tering and dynamic stress concentrations in the plates with two holes are studied by the refined dynamic equation of plate bending. The problem to be solved is changed to a set of infinite algebraic equations by an orthogonM function expansion method. As examples, under free boundary conditions, the numerical results of the dynamic moment concen- tration factors in the plates with two circular holes are computed. The results indicate that the parameters such as the incident wave number, the thickness of plates, and the spacing between holes have great effects on the dynamic stress distributions. The results are accurate because the refined equation is derived without any engineering hypothese.
文摘In this paper, based on complex variables and conformal mapping methods, using the refined dynamic equation of plates, elastic wave scattering and dynamic stress concentrations in plates with two cutouts were studied. Applying the orthogonal function expansion method, the problem to be solved can be reduced into the solution of a set of infinite algebraic equations. According to free boundary conditions, numerical results of dynamic moment concentration factors in thick plates with two circular cutouts analyze that: there will be more complex interaction changes between two-cutout situation than single cutout situation. In the case of low frequency or high frequency and thin plate, the hole-spacing in the absence of coupling interactions was larger or smaller. The numerical results and method can be used to analyze the dynamics and strength of plate-like structures.
基金supported in part Professor Yuesheng Xu under the program of"One Hundred Outstanding Young Chinese Scientists" of the Chinese Academy of Sciencesthe Graduate Innovation Foundation of the Chinese Academy of Sciences
文摘This paper is concerned with seeking the general solutions of matrix equation M（ξ）M＊ （ξ） = Is for the construction of multiple channel biorthogonal wavelets, provided that some special solution of its is known.
基金supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. LQ17E050011)the National Natural Science Foundation of China (Grant No. 51775154)+1 种基金the Natural Science Foundation of Zhejiang Province of China (Grant No.LQ17E090007)the Key Project of Natural Science Foundation of Zhejiang Province of China (Grant No. LQ17E050011)
文摘Based on the refined dynamic equation of stretching plates, the elastic tensio compression wave scattering and dynamic stress concentrations in the thick plate with two cutouts are studied. In view of the problem that the shear stress is automatically satisfied under the free boundary condition, the generalized stress of the first-order vanishing moment of shear stress is considered. The numerical results indicate that, as the cutout is thick, the maximum value of the dynamic stress factor obtained using the refined dynamic theory is 19% higher than that from the solution of plane stress problems of elastic dynamics.
