Convergence rates of cascade algorithms associated with nonhomogeneous refinement equations

This paper is concerned with nonhomogeneous refinement equations of the form

     

Convergence rates of cascade algorithms

We consider solutions of a refinement equation of the form

where is a finitely supported sequence called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the cascade algorithm associated with , i.e., the convergence of the sequence in the -norm.

Our main result gives estimates for the convergence rate of the cascade algorithm. Let be the normalized solution of the above refinement equation with the dilation matrix being isotropic. Suppose lies in the Lipschitz space , where 0$"> and . Under appropriate conditions on , the following estimate will be established:

where and is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.

     

Convergence of cascade algorithm for individual initial function and arbitrary refinement masks

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For any initial function φn, a cascade sequence (φn)n∞=1 constructed by the iteration φn=Cnφn-1=1,2.. where Cαis defined by g∈Lp(R) In this paper, we characterize the convergence of a cascade sequence in terms of a sequence of functions and in terms of joint spectral radius. As a consequence, it is proved that any convergent cascade sequence has a convergence rate of geometry, i.e., ||φ 1-φn||Lp(R)=O((？)n)for some (？)∈(0.1i). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.     

Stochastic nonhomogeneous incompressible Navier-Stokes equations

We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier-Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331-332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240-1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier-Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992].The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier-Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier-Stokes equations, Applicandae Math. 25 (1991) 59-85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Global L~2 Stability of the Nonhomogeneous Incompressible Navier–Stokes Equations

In this paper, the problem of the global L^2 stability for large solutions to the nonhomogeneous incompressible Navier-Stokes equations in 3D bounded or unbounded domains is studied. By delicate energy estimates and under the suitable condition of the large solutions, it shows that if the initial data are small perturbation on those of the known strong solutions, the large solutions are stable.     

Multivariate nonhomogeneous refinement equations

We give necessary and sufficient conditions for the existence and uniqueness of compactly supported distribution solutionsf=(f ₁,...,f _r)^T of nonhomogeneous refinement equations of the form

Vector refinement equations with infinitely supported masks

In this paper we investigate the L₂-solutions of vector refinement equations with exponentially decaying masks and a general dilation matrix. A vector refinement equation with a general dilation matrix and exponentially decaying masks is of the form

where the vector of functions φ=(φ₁,…,φ_r)^T is in is an exponentially decaying sequence of r×r matrices called refinement mask and M is an s×s integer matrix such that lim_n→∞M^-n=0. Associated with the mask a and dilation matrix M is a linear operator Q_a on given by

The iterative scheme is called vector subdivision scheme or vector cascade algorithm. The purpose of this paper is to provide a necessary and sufficient condition to guarantee the sequence to converge in L₂-norm. As an application, we also characterize biorthogonal multiple refinable functions, which extends some main results in [B. Han, R.Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, Appl. Comput. Harmon. Anal., to appear] and [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, Advances in Wavelet (Hong Kong, 1997), Springer, Singapore, 1998, pp. 199–227] to the general setting.     

Solutions of multiple vector refinement equations with infinite mask

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.     

Vector cascade algorithms with infinitely supported masks in weighted L 2-spaces

In this paper, we shall study the solutions of functional equations of the form Φ =∑α∈Zsa(α)Φ(M·-α), where Φ = (φ1, . . . , φr)T is an r×1 column vector of functions on the s-dimensional Euclidean space, a:=(a(α))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 are interested in the question, for a mask a with exponential decay, if there exists a solution Φ 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 Qa,Mf:=∑α∈Zsa(α)f (M·-α),f= (f1, . . . , fr)T∈(L2,μ(Rs))r.The iterative scheme (Qan,Mf)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2,μ(Rs))r , the weighted L2 space. Inspired by some ideas in [Jia,R.Q.,Li,S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008-1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L2(Rs))r, then its limit function belongs to (L2,μ(Rs))r for some μ0.     

Continuous refinement equations and subdivision

This paper is concerned with the study of a general class of functional equations covering as special cases the relation which defines theup-function as well as equations which arise in multiresolution analysis for wavelet construction. We discuss various basic properties of solutions to these functional equations such as regularity, polynomial containment within the space spanned by their integer shifts and their computability by subdivision algorithms.     

Convergence of cascade sequence on the Heisenberg group

The investigation of convergence of cascade sequence plays an important role in wavelet analysis on the Euclidean space and also in wavelet analysis on the Heisenberg group. This paper characterizes the -convergence of cascade sequence on the Heisenberg group in terms of the -norm joint spectral radius of a collection of matrices associated with the refinement sequence and gives a sufficient condition.

     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (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 W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (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∈L_p(ℝ^s) (φ∈C(ℝ^s) 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,L_p(ℝ^s)) 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.     

Convergence of nonstationary cascade algorithms

Summary. A nonstationary multiresolution of is generated by a sequence of scaling functions We consider that is the solution of the nonstationary refinement equations where is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in of the corresponding nonstationary cascade algorithm as k or n tends to It is assumed that there is a stationary refinement equation at with filter sequence h and that The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h. Received September 19, 1997 / Revised version received May 22, 1998 / Published online August 19, 1999     

Sobolev空间中与非齐次细分方程相关的细分格式的收敛阶

研究如下形式的非齐次细分方程,其中向量值函数是未知的,g是给定的紧支集向量值函数,a 是一个具有有限长的r×r矩阵值序列,称为细分面具,M是一个s×s整数矩阵, 并且满足limn→∞M-n=0.我们在Sobolev空间(Wpk(Rs))r(1≤p≤∞)中研究与非齐次细分方程相关的细分格式的收敛性和收敛阶.选择具有紧支集向量值函数,定义n=1,2,….这个叠代过程称为细分格式(详见文献[1-29]).     

Convergence analysis of the adaptive finite element method with the red-green refinement

In this paper, we analyze the convergence of the adaptive conforming P 1 element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the...     

On global solutions of gravity driven nonhomogeneous viscous flows in a bounded domain

We consider an initial boundary value problem for nonhomogeneous Navier‐Stokes equations with a uniform gravitational field. For any given steady density profile whose derivatives are sufficiently close to a negative constant, we show that there exists a unique global solution if the initial perturbation with respect to the steady state is sufficiently small.     

Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban–Lions domains with holes

In this article, we consider the following eigenvalue problems

('∗

λ' render=n">

where λ>0, N2 and is the upper semi-strip domain with a hole in . Under some suitable conditions on f and h, we show that there exists a positive constant λ^* such that Eq. (*)_λ has at least two solutions if λ(0,λ^*), a unique positive solution if λ=λ^*, and no positive solution if λ>λ^*. We also obtain some further properties of the positive solutions of (*)_λ.     

Convergence of cascade algorithms by frequency approach

Starting with an initial function ?₀, the cascade algorithm generates a sequence by cascade operator Qa defined by