LINEAR INDEPENDENCE OF THE INTEGER TRANS-LATES OF COMPACTLY SUPPORTED DISTRIBU-TIONS AND REFINABLE VECTORS

In this paper,the global and local linear independence of any compactly supported distributions by using time domain spaces,and of refinable vectors by invariant linear spaces are investigated.

CONVERGENCE RATE OF VECTOR SUBDIVISION SCHEME

In this paper, some characterizations on the convergence rate of both the homogeneous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.

Some theoretical comparisons of refined Ritz vectors and Ritz vectors

Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ,x)of a large matrix A. Given a subspace w that contains anapproximation to x, these two methods compute approximations(μ(x~)) and (μ(x^)) to (λ,x),respectively. We establish three results. First, the refinedeigenvector approximation or simply the refined Ritz vector (x^) is unique as the deviation of x from w approaches zero if λ is simple. Second, interms of residual norm of the refined approximate eigenpair (μ,(x^)), we derive lower and upper bounds for the sine of the angle betweenthe Ritz vector (x~) and the refined eigenvector approximation (x^), and we prove that (x~)≠(x^) unless (x^)=x. Third, we establish relationships between theresidual norm ‖A(x~)-μ(x^)‖ of the conventionalmethods and the residual norm ‖A(x^)-μ(x^)‖ of therefined methods, and we show that the latter is always smallerthan the former if (μ,(x^)) is not an exact eigenpair ofA, indicating that the refined projection method is superiorto the corresponding conventional counterpart.

Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions

Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.

The Support of a Refinable Vector Satisfying an Inhomogeneous Refinement Equation

In this paper, we investigate the support of a refinable vector satisfying an inhomoge- neous refinement equation. By using some methods introduced by So and Wang, an estimate is given for the support of each component function of a compactly supported refinable vector satisfying an inhomogeneous matrix refinement equation with finitely supported masks.

A NOTE ON VECTOR CASCADE ALGORITHM

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.