Inequalities for nonuniform wavelet frames

Abstract

Gabardo and Nashed have studied nonuniform wavelets based on the theory of spectral pairs for which the associated translation set Λ = {0, r/N } + 2 Z is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we construct the associated wavelet frames and establish some sufficient conditions that ensure the nonuniform wavelet system ψ j,λ (x) = (2N ) j/2 ψ (2N ) j x − λ , j ∈ Z, λ ∈ Λ to be a frame for L 2 (R). The conditions proposed are stated in terms of the Fourier transforms of the wavelet system's generating functions.

Fractional nonuniform multiresolution analysis in L2(ℝ)

“…These studies were continued by Gabardo and his colleagues in previous works, 3–5 wherein they derived an extension of Cohen's theorem which gives the necessary and sufficient condition for the orthonormality of the collection and provided a complete characterization of associated wavelets by means of its dimension function. The theory of nonuniform wavelets were further studied and investigated by several researchers in different directions, for instance, biorthogonal nonuniform wavelets, 6 characterization of scaling functions associated with an NUMRA, 7 nonuniform wavelet packets, 8 nonuniform wavelet frames (see Dai et al 9 and Debnath 10 ), nonuniform wavelets and wavelet packets on local fields of positive characteristic (see previous works 11–13 ) and vector‐valued nonuniform wavelets and wavelet packets (see previous works 14,15 ).…”

Nonuniform Multiresolution Analysis Associated with Linear Canonical Transform

“…These studies were proceeded by Gabardo and his colleagues in [10,11,28], wherein they establish an extension of Cohen's theorem which provides the necessary and sufficient condition for the orthonormality of the system {φ(• − λ) : λ ∈ Ω} and presented some equivalent conditions of the associated wavelets via dimension functions. The theory of nonuniform wavelets was further studied and extensively investigated by many researchers in different directions, for example, nonuniform wavelet packets [2], vector-valued nonuniform wavelet packets [15], generalized nonuniform MRA [16], nonuniform wavelet frames [18,23], nonuniform wavelets and wavelet packets on local fields of positive characteristic [19,20,21].…”

AB-wavelet frames in L2(Rn)

“…For instance, Daubechies [3] proved the first result on the necessary and sufficient conditions for the following conventional wavelet system ψ j,k := a j/2 ψ a j x−kb : j, k ∈ Z to constitute a frame for L 2 (R n ), Chui and Shi [2] refined the result of Daubechies in [3], Christenson [1] established a stronger version of Daubechies sufficient condition for wavelet frames. Recently, these conditions have been further refined and investigated by several authors (see, for example, [10][11][12][13]15]). Therefore, the main objective of this article is to establish conditions on the wavelet function ψ and the dilation and translation parameters so that the corresponding AB-wavelet system W AB (ψ, j, k) given by (1.1) constitutes a frame for L 2 (R n ).…”