Stability and linear independence associated with wavelet decompositions

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 satisfied by the basis function.

Dependence Relations Among the Shifts of a Multivariate Refinable Distribution

“…In the univariate case, a useful characterization of linear independence for the shifts of a single function φ was given in terms of a by Jia and Wang in [17]. Related results were provided by Ron in [23] and Zhou in [28].…”

“…Attempts to generalize their results to functions of several variables have been made by Hogan [12] and Zhou [30]. In [12], the conditions of [17] were shown to be necessary in several variables; and they were shown to be also sufficient for functions of a certain type. These results were not satisfactory for two reasons: the proofs do not apply to general multivariate functions; and the conditions, though easy to verify in the univariate case, are more elusive in several variables.…”

Linear independence of pseudo-splines

“…We start with two lemmata. The first lemma is implied by Theorem 1 and 2 of a paper of Jia and Wang (see [16]). Instead of stating both theorems of [16] and deducing the following lemma by using them, here we include a direct proof which is essentially derived from Jia and Wang's proof of Theorem 1 and 2 in [16].…”

“…Since ( φ(2kπ)) k∈Z = 0, ζ 0 ∈ C \ {0}. Applying (1.2) again, one obtains, for any k ∈ Z, [19], the set A must be finite (also see [16]). Hence, there must exist some integers 0…”

“…In [19] Ron (also see [4]) studied the linear independence of compactly supported distributions in terms of their Fourier-Laplace transforms. Applying Proposition 2.1 of [19] (also see [16]), one obtains that for an arbitrary compactly supported single variable distribution, which is not identically zero, there are at most finitely many ζ ∈ C such that…”