Norms Concerning Subdivision Sequences and Their Applications in Wavelets

We study multivariate trigonometric polynomials satisfying the "sum-rule" conditions of a certain order. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest in the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. The approach presented in this paper leads directly to constructive algorithms, and is an alternative to the analysis of multivariate subdivision schemes in terms of the joint spectral radius of certain operators. Our convergence results apply to arbitrary dilation matrices, while the smoothness results are limited to two classes of dilation matrices.

“…The latter approach, is presented, in particular, in the papers [1,2,4,7,9,10,18]. These two approaches are shown to be equivalent in the recent paper [3].…”

Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 91%

Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes

Dyn

Skopina

2011

“…Smoothness property of refinable function vectors under various conditions has been extensively investigated in the literature, see [1,4,5,7,9,11,16,17,21,26,27,29,31] and many references therein.…”

Section: Characterization Of Smoothness Of Refinable Function Vectorsmentioning

confidence: 99%

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

Han

2006

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W k p (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 k p (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.

“…Subdivision schemes have been studied mainly for the case in which the mask a is finitely supported (see [1,2,7,9,14,15,17,18,21,23,24,28,31]). However, in some applications of signal proceeding, one needs to find solutions of refinement equation (1.1) with infinitely supported masks, such as the band limited wavelets which were generated by refinement equations (1.1) with infinitely supported masks [12,13,25,30].…”

Section: Introductionmentioning

confidence: 99%

Subdivision schemes with polynomially decaying masks

Pan

2009

In this paper we investigate the L 2 -solutions of vector refinement equations with polynomially decaying masks and a general dilation matrix, which plays a vital role for characterizations of wavelets and biorthogonal wavelets with infinite support. A vector refinement equation with polynomially decaying masks and a general dilation matrix is the form:where the vector of functions φ = (φ 1 , · · · , φ r ) T is in (L 2 (R s )) r , a := (a(α)) α∈Z s is a polynomially decaying sequence of r × r matrices called refinement mask and M is an s × s integer matrix such that lim n→∞ M −n = 0. The corresponding cascade operator on (L 2 (R s )) r is given by:The iterative scheme (Q n a f ) n=1,2,··· , is called vector cascade algorithm. In this paper we give a complete characterization of convergence of the sequence (Q n a f ) n=1,2··· in L 2 -norm. Some properties of the transition operator restricted to a certain linear space are discussed. As an application of convergence, we also obtain a characterization of smoothness of solutions of refinement equation mentioned above for the case r = 1.