Stability and Linear Independence Associated with Wavelet Decompositions

Abstract

Abstract.Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence, and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation that the basis function satisfies.

Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems

“…Thus, the stability of a compactly supported function φ ∈ L 2 (R) is equivalent to (3.2). The linear independence of the integer translates of a refinable function φ is characterized in terms of their masks [16]. The main results, Theorem 1 and 2 in [16], imply directly the following lemma.…”

“…The linear independence of the integer translates of a refinable function φ is characterized in terms of their masks [16]. The main results, Theorem 1 and 2 in [16], imply directly the following lemma. Here, the notion of symmetric zeros is used: A Laurent polynomial a(z) has a pair of symmetric zeros on…”

Complex Wavelets and Framelets from Pseudo Splines

“…It was shown in [19,28] Proof Let 2 φ be the pseudo spline of type II. It was shown in [9] that the integer shifts of 2 φ are linearly independent.…”

Local polynomial property and linear independence of refinable distributions

“…The local linear independence plays an important role in spline interpolation as well as in nonlinear wavelet approximation (see [4,6] and references therein). About the global and local linear independence of compactly supported refinable distributions, there is a long list of publications (see for instance [2,6,8,9,11,12]). For a refinable distribution, its global linear independence is characterized by corresponding symbol in [8], but its local linear independence is not easy to be checked in general.…”

“…About the global and local linear independence of compactly supported refinable distributions, there is a long list of publications (see for instance [2,6,8,9,11,12]). For a refinable distribution, its global linear independence is characterized by corresponding symbol in [8], but its local linear independence is not easy to be checked in general. When M = 2, it was proved that the local and global linear independence of refinable distributions are equivalent to each other (see [9] for refinable functions with biorthogonal dual and [11] for any refinable distributions).…”