Matrix Fourier multipliers for Parseval multi-wavelet frames (original) (raw)

Parseval frame wavelets with E n ( 2 ) -dilations

Applied and Computational Harmonic …, 2005

We study Parseval frame wavelets in L 2 (R n ) with matrix dilations of the form (Df )(x) = √ 2f (Ax), where A is an arbitrary expanding n × n matrix with integer coefficients, such that |det A| = 2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low pass filters.

Parseval frame wavelets with E ( 2 ) n-dilations ✩

2005

We study Parseval frame wavelets in L2(Rn) with matrix dilations of the form(Df )(x) = √2f (Ax), whereA is an arbitrary expanding n × n matrix with integer coefficients, such that |detA| = 2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators seval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval fra (bi)wavelets associated with A-MRA’s are described. We then introduce new classes of filter induced wavele bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelet and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseva (bi)wavelets from generalized low-pass filters.  2005 Elsevier Inc. All rights reserved.

Parseval frame wavelets with -dilations

2005

Infinite matrices, wavelet coefficients and frames

2004

We study the action of A on f ∈ L 2 (R) and on its wavelet coefficients, where A = (a lmjk ) lmjk is a double infinite matrix. We find the frame condition for A-transform of f ∈ L 2 (R) whose wavelet series expansion is known.

Wavelet Multipliers on L(R)

2007

We give results on the boundedness and compactness of wavelet multipliers on Lp(Rn), 1 ≤ p ≤ ∞. 1. Wavelet multipliers Let σ ∈ L∞(Rn). Then we define the linear operator Tσ : L(R) → L(R) by Tσu = (σû)∨, u ∈ L(R), where û is the Fourier transform of u defined by û(ξ) = lim R→∞ (2π)−n/2 ∫ |x|≤R e−ix·ξu(x) dx, ξ ∈ R, the convergence is understood to take place in L(R) and (σû)∨ is the inverse Fourier transform of σû. It is a consequence of Plancherel’s theorem that Tσ : L(R) → L(R) is a bounded linear operator. Let π : R → U(L(R)) be the unitary representation of the additive group R on L(R) defined by (π(ξ)u)(x) = eix·ξu(x), x, ξ ∈ R, for all functions u in L(R), where U(L(R)) is the group of all unitary operators on L(R). Let φ be any function in L(R) ∩ L∞(Rn) such that ‖φ‖2 = 1, where ‖ ‖p denotes the norm in L(R) for 1 ≤ p ≤ ∞. Then it is proved in [9] that (1.1) (φu, φv) = (2π)−n ∫ Rn (u, π(ξ)φ)(π(ξ)φ, v) dξ for all functions u and v in the Schwartz space S, where ( , ) is the inn...

The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames

Canadian Mathematical Bulletin, 2013

In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as π(Γ)ψ, where π is a unitary representation of a wavelet group and Γ is the abstract pseudo-lattice Γ. We prove a condition in order that a Parseval frame π(Γ)ψ can be dilated to an orthonormal basis of the form τ (Γ)Ψ where τ is a super-representation of π. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.

Orthonormal dilations of Parseval wavelets

Mathematische Annalen, 2008

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group

Construction of Parseval wavelets from redundant filter systems

Journal of Mathematical Physics, 2005

We consider wavelets in L 2 (R d ) which have generalized multiresolutions. This means that the initial resolution subspace V 0 in L 2 (R d ) is not singly generated. As a result, the representation of the integer lattice Z d restricted to V 0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on R d can be constructed directly from the generalized wavelet filters.

Orthogonal wavelet frames and vector-valued wavelet transforms

Applied and Computational Harmonic Analysis, 2007

Motivated by the notion of orthogonal frames, we describe sufficient conditions for the construction of orthogonal MRA wavelet frames in L 2 (R) from a suitable scaling function. These constructions naturally lead to filter banks in 2 (Z) with similar orthogonality relations and, through these filter banks, the orthogonal wavelet frames give rise to a vector-valued discrete wavelet transform (VDWT). The novelty of these constructions lies in their potential for use with vector-valued data, where the VDWT seeks to exploit correlation between channels. Extensions to higher dimensions are natural and the constructions corresponding to the bidimensional case are presented along with preliminary results of numerical experiments in which the VDWT is applied to color image data.

The dilation property for abstract Parseval wavelet systems

In this work we introduce a class of discrete groups called wavelet groups that are generated by a discrete group Γ0 (translations) and a cyclic group Γ1 (dilations), and whose unitary representations naturally give rise to a wide variety of wavelet systems generated by the pseudo-lattice Γ = Γ1Γ0. We prove a condition in order that a Parseval frame wavelet system generated by Γ can be dilated to an orthonormal basis that is also generated by Γ via a super-representation. For a subclass of groups where Γ0 is Heisenberg, we show that this condition always holds, and we cite a number of familiar examples as applications.

Wavelets and Linear Algebra 3 ( 2 ) ( 2016 ) 1932

2016

In this paper we study necessary and sufficient conditions for some types of linear combinations of wave packet frames to be a frame for L 2 (R d). Further, we illustrate our results with some examples and applications.

Construction of Wavelet and Gabor's Parseval Frames

A new way to build wavelet and Gabor's Parseval frames for L 2 (R d ) is shown in this paper. In the first case the construction is done using an expansive matrix B, together with only one function h ∈ L 2 (R d ). In the second one, we work with a function g ∈ L 2 (R d ) and two invertible matrixes B and C, with the condition that C t Z d ⊂ Z d . The only requirement for h and g is that they have to be supported in a set Q, such that the measure of Q is finite and positive. Q has diameter lower than 1, and its border has null measurement. In addition,

Construction of orthonormal multi-wavelets with additional vanishing moments

Advances in Computational Mathematics, 2006

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ = [φ1, • • • , φr] be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ = [ψ1, • • • , ψr] an o.n. multiwavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r + 1)-dimensional o.n. scaling function vector φ := [φ , φr+1] and some corresponding o.n. multi-wavelet ψ are constructed in such a way that φ has p.p.o. = n > m and their two-scale symbols P and Q are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r = 1, if we consider the m th order Daubechies o.n. scaling function φ D m , then φ := [φ D m , φ2] is a scaling function vector with p.p.o. > m. As another example, for r = 2, if we use the symmetric o.n. scaling function vector φ in our earlier work [3], then we obtain a new pair of scaling function vector φ = [φ , φ3] and multiwavelet ψ that not only increase the order of vanishing moments but also preserve symmetry.

Multipliers, Phases and Connectivity of MRA Wavelets in L 2(ℝ2)

Journal of Fourier Analysis and Applications, 2010

Let A be any 2 × 2 real expansive matrix. For any A-dilation wavelet ψ, let ψ be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of (f ψ) is an A-dilation wavelet for any A-dilation wavelet ψ. In this paper, we give a complete characterization of all A-dilation wavelet multipliers under the condition that A is a 2 × 2 matrix with integer entries and |det(A)| = 2. Using this result, we are able to characterize the phases of A-dilation wavelets and prove that the set of all A-dilation MRA wavelets is path-connected under the L 2 (R 2 ) norm topology for any such matrix A.

Frame wavelets in subspaces of L2(mathbbRd)L^2(\mathbb R^d)L2(mathbbRd)

Proceedings of the American Mathematical Society, 2002

Let A be a d × d real expansive matrix. We characterize the reducing subspaces of L 2 (R d) for A-dilation and the regular translation operators acting on L 2 (R d). We also characterize the Lebesgue measurable subsets E of R d such that the function defined by inverse Fourier transform of [1/(2π) d/2 ]χ E generates through the same A-dilation and the regular translation operators a normalized tight frame for a given reducing subspace. We prove that in each reducing subspace, the set of all such functions is nonempty and is also path connected in the regular L 2 (R d)-norm.

A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of 𝐿²(ℝⁿ), where 𝕟≥2

Electronic Research Announcements of the American Mathematical Society, 2003

In this note, we announce a general method for the construction of nonseparable orthogonal wavelet bases of L 2 ( R n ) , L^2(\mathbb {R}^n), where n ≥ 2. n\geq 2. Hence, we prove the existence of such type of wavelet bases for any integer n ≥ 2. n\geq 2. Moreover, we show that this construction method can be extended to the construction of n n -D multiwavelet matrix filters.

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

Proceedings of the American Mathematical Society, 2004

We introduce the concept of the modular function for a shiftinvariant 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 multiresolution 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.

Construction of Non-Uniform Parseval Wavelet Frames for L2 (R) via UEP

2019 13th International conference on Sampling Theory and Applications (SampTA), 2019

We study the construction of non-uniform Parseval wavelet frames for the Lebesgue space L2 (R), where the related translation set is not necessary a group. The unitary extension principle (UEP) and generalized (or oblique) extension principle (OEP) for the construction of multi-generated non-uniform Parseval wavelet frames for L2 (R) are discussed. Some examples are also given to illustrate our results.

An Equivalence Relation on Wavelets in Higher Dimensions

Bulletin of the London Mathematical Society, 2004

We introduce an equivalence relation on the set of single wavelets of L 2 (R n) associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of wavelets and it is shown that each of these classes is non-empty. 2000 Mathematics Subject Classification. 42C40.