On refinable functions and subdivision with positive masks
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Springer 2006
On refinable functions and subdivision with positive masks Johan De Villiers Department of Mathematics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa E-mail: [email protected]
Received 23 April 2003; accepted 12 March 2004 Communicated by T. Sauer
Dedicated to Charles Micchelli on his sixtieth birthday
We consider aspects of the analysis of refinement equations with positive mask coefficients. First we derive, explicitly in terms of the mask, estimates for the geometric convergence rate of both the cascade algorithm and the corresponding subdivision scheme, as well as the Hölder continuity exponent of the resulting refinable function. Moreover, we show that the subdivision scheme converges for a class of unbounded initial sequences. Finally, we present a regularity result containing sufficient conditions on the mask for the refinable function to possess continuous derivatives up to a given order. Keywords: refinable functions, subdivision, wavelets, cascade algorithm. Mathematics subject classifications (2000): 42C40, 65T60.
1.
Introduction
We write M(Z) and M(R) for, respectively, the bi-infinite real-valued sequences and the real-valued functions on R. The sequences and functions of finite support are denoted respectively by M0 (Z) and M0 (R). Similarly, the symbol C0 (R) is used for the space of continuous functions (on R) with finite support. The backward difference operator : M(Z) → M(Z) is defined, for a sequence c = {cj : j ∈ Z} ∈ M(Z), by (c)j = cj − cj −1 , j ∈ Z, and we shall write ∞ (Z) for the subspace of M(Z) consisting of those bi-infinite sequences c which are such that c ∈ ∞ (Z). Observe that ∞ (Z) is a proper subspace of ∞ (Z). The set of non-negative integers will be denoted by Z+ . A function φ ∈ M(R) is called refinable with respect to a mask a ∈ M0 (Z) if and only if the refinement equation aj φ(2x − j ), x ∈ R, (1.1) φ(x) = j
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is satisfied, and where we have adopted the notation j := j ∈Z . The corresponding mask symbol A is then defined as the Laurent polynomial aj zj , z ∈ C \ {0}. (1.2) A(z) = j
The study of the existence, uniqueness and other relevant properties of refinable functions plays a fundamental role in both subdivision (see [1;9, theorem 2.5]) and wavelet analysis (see, e.g., [2,3;9, section 2.6]). For a given mask a ∈ M0 (Z), recall that the subdivision operator Sa : M(Z) → M(Z) is defined by aj −2k ck , j ∈ Z. (1.3) (Sa c)j = k
The corresponding subdivision scheme then generates, for a given initial sequence c ∈ M(Z), the sequence {c(r) : r ∈ Z+ } ⊂ M(Z) as defined recursively by c(0) = c,
c(r+1) = Sa c(r) ,
r ∈ Z+ .
(1.4)
It should be pointed out that the subdivision results quoted and proved in this paper can all easily be extended to hold also for vector-valued initial sequences c in (1.4). For simplicity, we shall not consider this complication. In this paper, we are primarily concerned with masks a ∈ M0 (Z) which, for a given integer n 2, sa
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