An algorithm for constructing multidimensional biorthogonal periodic multiwavelets

Goh, Say Song; Teo, Kok-Ming

doi:10.1017/s0013091500021246

Cited by 8 publications

(4 citation statements)

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“…As shown in [6], (2.6) is equivalent to biorthogonality of {T ℓ φ m } m=1,...,r;ℓ∈R and {T ℓφ m } m=1,...,r;ℓ∈R . It then follows that the functions in {T ℓ φ m } m=1,...,r;ℓ∈R are linearly independent.…”

Section: Oblique Duals Of Finitely-generated Framesmentioning

confidence: 99%

“…The theory of periodic wavelets via polyphase splines in [5,6,7] was developed for the more general multidimensional setting of L 2 ((0, 2π) s ), where s ∈ N, as minimal additional efforts compared to the one-dimensional case were needed. Likewise, all the results here on oblique duals for finite-dimensional spaces in L 2 (0, 2π) could be easily extended to the multidimensional setting.…”

Section: Oblique Duals In Prescribed Subspacesmentioning

confidence: 99%

“…, r; ℓ ∈ R}; (1.1) the underlying multiresolution analysis. The multiple generator setup of (1.1) reflects the situations encountered with multiple refinable functions, multiwavelets and wavelet frames; see, e.g., [5,6,7]. Our purpose here is to obtain expansions of the functions in V of the form…”

Section: Introductionmentioning

confidence: 99%

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Pairs of oblique duals in spaces of periodic functions

Christensen

Goh

2009

Adv Comput Math

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Abstract. We construct non-tight frames in finite-dimensional spaces consisting of periodic functions. In order for these frames to be useful in practice one needs to calculate a dual frame; while the canonical dual frame might be cumbersome to work with, the setup presented here enables us to obtain explicit constructions of some particularly convenient oblique duals. We also provide explicit oblique duals belonging to prescribed spaces different from the space where we obtain the expansion. In particular this leads to oblique duals that are trigonometric polynomials.

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Section: Oblique Duals Of Finitely-generated Framesmentioning

confidence: 99%

Section: Oblique Duals In Prescribed Subspacesmentioning

confidence: 99%

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Pairs of oblique duals in spaces of periodic functions

Christensen

Goh

2009

Adv Comput Math

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“…Later, trigonometric scaling functions and wavelets were constructed directly from a PMRA on the unit circle [8,9,12,15,34]. More general constructions of periodic orthogonal wavelets or biorthogonal multidimensional multiwavelets emerged from non-stationary PMRA's in the sense that different scaling functions and wavelets are involved at different scales [22,27]. Constructions of tight periodic (multi)wavelet frames were obtained in [20,21,23,25,33].…”

Section: Introductionmentioning

confidence: 99%

Characterizations of dual multiwavelet frames of periodic functions

Atreas

2016

Int. J. Wavelets Multiresolut Inf. Process.

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We prove characterizations of dual multiwavelet frames for [Formula: see text] constructed from a pair of [Formula: see text]-periodic multirefinable generators. Based on the notion of the Mixed Fundamental sequence we derive Mixed Extension Principles dictating the construction of dual wavelet masks with respect to given refinement masks and we show that these Principles characterize dual framelets.

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Wavelet frames and shift-invariant subspaces of periodic functions

Goh

Teo

2006

Applied and Computational Harmonic Analysis

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We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix A and the multiplicity functions for general multires-olution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters. 1. Preliminaries A frame for a separable Hilbert space H is a sequence of vectors {f j } in H such that there exist constants C 1 , C 2 > 0 such that C 1 f 2 2 ≤ j ||f, f j | 2 ≤ C 2 f 2 2 holds for all f ∈ H. If C 1 = C 2 = 1, we say that {f j } is a normalized tight frame. For a d × d real expansive matrix A (i.e., all the eigenvalues of A are required to have absolute values greater than 1), an A-dilation (orthogonal) wavelet is a single function ψ ∈ L 2 (R d) with the property that ψ A m,, (t) := {|detA| m 2 ψ(A m t −) : m ∈ Z, , ∈ Z d } is an orthonormal basis for L 2 (R d). More generally, ψ will be called a normalized tight A-dilation wavelet frame if {ψ A m,, } forms a normalized tight frame for L 2 (R d). In what follows we will use the term "wavelet frame" to denote the normalized tight ones. The dilation operator δ A and the translation operator T are defined by: (δ A f)(t) = |detA| 1/2 f (At), (T f)(t) = f (t −)

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An algorithm for constructing multidimensional biorthogonal periodic multiwavelets

Cited by 8 publications

References 10 publications

Pairs of oblique duals in spaces of periodic functions

Pairs of oblique duals in spaces of periodic functions

Characterizations of dual multiwavelet frames of periodic functions

Wavelet frames and shift-invariant subspaces of periodic functions

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