On the Construction of Multivariate (pre) Wavelets

Abstract

A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shift-invariant spaces and orthogonal projectors from L 2 (lRd) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution.

The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

“…An alternative proof is found in [18], and, in either case, we have the following result. THEOREM 4.5.…”

Characterization of low pass filters in a multiresolution analysis

“…Thus if we take ϕ = φ h (Φ φ h ) −1/2 , where Φ φ h is defined by (1), then ϕ will be a scaling function of an A-MRA (see [1,Section 2]) and the following claim is true:…”

“…Properties of scaling functions have been studied by several authors (see [20], [15], [8], [10], [1], [7], [13], [18], [4]). …”

On Multivariate Compactly Supported Bi-Frames

“…Due to [11], if additionally ϕ ∈ L 2 (R d ) and ϕ(0) = 0, then it generates a multiresolution analysis, i.e., an increasing sequence (V j ) j ∈Z of closed subspaces in L 2 (R d ) such that…”