Hermite B-Splines: n-Refinability and Mask Factorization

This paper deals with polynomial Hermite splines. In the first part, we provide a simple and fast procedure to compute the refinement mask of the Hermite B-splines of any order and in the case of a general scaling factor. Our procedure is solely derived from the polynomial reproduction properties satisfied by Hermite splines and it does not require the explicit construction or evaluation of the basis functions. The second part of the paper discusses the factorization properties of the Hermite B-spline masks in terms of the augmented Taylor operator, which is shown to be the minimal annihilator for the space of discrete monomial Hermite sequences of a fixed degree. All our results can be of use, in particular, in the context of Hermite subdivision schemes and multi-wavelets.

Vector-valued nonuniform multiresolution analysis on local fields

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

Convergence Rates of Cascade Algorithms with Infinitely Supported Masks

We investigate the solutions of refinement equations of the formwhere the function ϕ is in Lp(ℝs)(1 ≤ p ≤ ∞), a is an infinitely supported sequence on ℤs called a refinement mask, and M is an s × s integer matrix such that limn→1M–n = 0. Associated with the mask a and M is a linear operator Qa,M defined on Lp(ℝs) by Qa,Mϕ0 := Σα∈ℤsa(α)ϕ0(M · –α). Main results of this paper are related to the convergence rates of in Lp(ℝs) with mask a being infinitely supported. It is proved that under some appropriate conditions on the initial function ϕ0, converges in Lp(ℝs) with an exponential rate.

Local linear independence of refinable vectors of functions

This paper is devoted to a study of local linear independence of refinable vectors of functions. A vector of functions is said to be refinable if it satisfies the vector refinement equation where a is a finitely supported sequence of r × r matrices called the refinement mask. A complete characterization for the local linear independence of the shifts of ϕ1,…,ϕr is given strictly in terms of the mask. Several examples are provided to illustrate the general theory. This investigation is important for construction of wavelets on bounded domains and nonlinear approximation by wavelets.

Approximation by Multiple Refinable Functions

We consider the shift-invariant space, 𝕊(Φ), generated by a set Φ = {Φ1,..., Φr} of compactly supported distributions on R when the vector of distributions ϕ:= {Φ1,..., Φr} T satisfies a system of refinement equations expressed in matrix form aswhere a is a finitely supported sequence of r x r matrices of complex numbers. Such multiple refinable functions occur naturally in the study of multiple wavelets.The purpose of the present paper is to characterize the accuracy of Φ, the order of the polynomial space contained in 𝕊(Φ), strictly in terms of the refinement mask a. The accuracy determines the Lp-approximation order of 𝕊(Φ) when the functions in (Φ) belong to Lp(ℝ) (see Jia [10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask a. In particular, they extend and improve the results of Heil, Strang and Strela [7], and of Plonka [16]. In addition, a counterexample is given to the statement of Strang and Strela [20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of Φ.