Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Abstract

In this work we prove that any pair of homogeneous dual multiwavelet frames of L 2 (R s ) constructed from a pair of refinable function vectors gives rise to a pair of nonhomogeneous dual multiwavelet frames and vice versa. We also prove that the Mixed Oblique Extension Principle characterizes dual multiwavelet frames. Our results extend recent characterizations of affine dual frames derived from scalar refinable functions obtained in [3].

Homogeneous wavelets and framelets with the refinable structure

“…We shall obtain almost complete satisfactory answers to this topic. In particular, our results on homogeneous framelets include all the results in [2,3,22] as special cases.…”

“…We now connect a homogeneous dual framelet with a nonhomogeneous dual framelet as follows. Under the condition that both Ψ andΨ are obtained from refinable functions through the refinable structure, item (i) of Theorem 4.4 is known in [2,3,22] for the existence of a dual M-framelet ({Φ;Ψ}, {Φ; Ψ}). Therefore, Theorem 4.4 generalizes [2,3,22] under a weaker assumption.…”

“…In Section 4, we link homogeneous dual framelets to nonhomogeneous dual framelets. Comparing with [2,3,22], results in Sections 3 and 4 on homogeneous framelets are much more general and complete. In the last Section 5, we discuss the relations between homogeneous wavelets and nonhomogeneous wavelets with the refinable structure.…”

“…Due to many desirable properties of nonhomogeneous affine systems, it is of interest in both theory and application to investigate when a homogeneous affine system can be linked to a nonhomogeneous affine system. Along this direction, [2,3] and [22,Section 4.5] have studied the connections of a homogeneous tight/dual framelet to a nonhomogeneous tight/dual framelet, while the connection of a homogeneous wavelet with a multiresolution analysis has been discussed in [6,22,26,30,31,37,38,47] and references therein. In this paper we shall comprehensively study the connections of homogeneous wavelets and framelets with nonhomogeneous wavelets and framelets and their connections to the refinable structure.…”

Characterizations of dual multiwavelet frames of periodic functions

“…Extension Principles were first proposed by Ron and Shen [29,30] and subsequently were extended by Daubechies et al in the form of the Oblique Extension Principle [14]. Extension Principles are used for the construction of affine dual frames of L 2 (R s ) arising from a pair of refinable functions or function vectors, see [1,2,4,5,14,29,30,31] and references therein. In [21], the authors derived a Unitary Extension Principle and a weak version of an Oblique Extension Principle for Parseval multiwavelet frames of type (1.5).…”