Subdivision schemes with polynomially decaying masks

Abstract

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.

Smoothness of multivariate refinable functions with infinitely supported masks

“…In addition, since a satisfies the basic sum rule if and only ifâ(2π(M T ) −1 ω) = 0, ∀ ω ∈ Ω \ {0}, the masks a N associated with φ N also satisfy the basic sum rule as soon as N is sufficiently large. By virtue of [30,Theorem 3.2], we conclude that the cascade algorithms associated with masks a N and dilation matrix M converge in L 2 (R s ), which immediately implies φ N ∈ L 2 (R s ) for all N large enough.…”

“…Since the shifts of φ are stable, by [30,Theorem 3.3], the cascade algorithm associated with mask a and dilation matrix M converges in L 2 (R s ). Thus, a satisfies the basic sum rule and…”

“…The space Y is important in the study of cascade algorithms with infinitely supported masks (see, e.g., [13,30,31]). …”

“…Recently, Han and Shen [14,15] characterized biorthogonal wavelets with polynomially decaying masks for the cases M = 2I s×s and M = 2. Li and Pan [30] also investigated the L 2 -solutions of vector refinement equations with polynomially decaying masks and a general dilation matrix.…”

Convergence Rates of Cascade Algorithms with Infinitely Supported Masks

“…However, due to some desirable properties, infinitely supported masks including masks with exponential decay such as Butterworth filters and masks with polynomial decay such as various types of fractional splines are also of interest in some applications in the area of digital signal processing ( [4,5,7,12,27]). More recently, the convergence of cascade algorithms associated with an infinitely supported mask has been investigated by some authors (see [9][10][11][21][22][23]).…”