Normalized Tight Frame Wavelet Sets in R d

Ill J Math, 2010

We show that for any d × d expansive matrix A with integer entries and | det(A)| = 2, the set of all A-dilation MRA wavelets is path-connected under the L 2 (R d) norm topology. We do this through the application of A-dilation wavelet multipliers, namely measurable functions f with the property that the inverse Fourier transform of (f ψ) is an A-dilation wavelet for any Adilation wavelet ψ (where ψ is the Fourier transform of ψ). In this process, we have completely characterized all A-dilation wavelet multipliers for any integral expansive matrix A with | det(A)| = 2.

2004

An s-elementary frame wavelet is a function ψ∈L 2 (R) which is a frame wavelet and is defined by a Lebesgue measurable set E⊂R such thatψ= 1 √ 2π χ E . In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the L 2 (R)-norm. This result also holds for s-elementary A-dilation frame wavelets in L 2 (R d ) in general. We also consider the uniform path-connectivity of the sets of frame wavelets, normalized tight frame wavelets and s-elementary frame wavelets. We prove that none of these sets is uniformly path-connected.

Eprint Arxiv Math 0703438, 2007

In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family of invertible dxd matrices {A_j in GL_d(R): j in J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L^2(R^d) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R^d. Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.

Appl Comput Harmonic Anal, 2004

In this article, we develop a general method for constructing wavelets {| det A j | 1/2 ψ(A j x − x j,k ) : j ∈ J, k ∈ K} on irregular lattices of the form X = {x j,k ∈ R d : j ∈ J, k ∈ K}, and with an arbitrary countable family of invertible d × d matrices {A j ∈ GL d (R) : j ∈ J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L 2 (R d ) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R d . Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.

2000

The paper studies orthonormal wavelets in L2(Rn) with dilations induced by expanding matrices with integer coe-cients of arbitrary deter- minant. We provide a method of construction of all scaling sets and, hence, of all orthonormal MSF wavelets with the additional property that the core space of the underlying multiresolution structure is singly generated. Several examples on the real line and

Journal of Computational and Applied Mathematics, 2003

Let H be a reducing subspace of L 2 (R d ), that is, a closed subspace of L 2 (R d ) with the property that

2004

In this paper, we will start the discussion with the refinable generators of the shift invariant (SI) spaces in L2(R) that possess the largest possible regularities and required vanishing moments. For the pseudo-scaling generators, the corresponding MRA frame wavelets with certain regularities are constructed. In addition, the stability of the refinable SI spaces and the corresponding complementary spaces, biorthogonality of the SI spaces, and the approximation property of the spaces are also discussed.

In this paper we employ recent results from real algebraic geometry and theory of moment problems to make the first step towards resolving the question of existence of multivariate tight wavelet frames whose generators have at least one vanishing moment. The so-called Unitary Extension Principle from [30] and the results in [28] allow us to reformulate the question of existence of tight wavelet frame in terms of the existence of the sum of squares decomposition of a single trigonometric polynomial with real coefficients. Our main result confirms the existence of such decompositions in the two-dimensional case. We also give sufficient conditions for existence of tight wavelet frames in the dimension d ≥ 3 and illustrate our results with several examples.

Constructive Approximation, 2013

In this paper we employ recent results from real algebraic geometry and theory of moment problems to make the first step towards resolving the question of existence of multivariate tight wavelet frames whose generators have at least one vanishing moment. The so-called Unitary Extension Principle from [30] and the results in [28] allow us to reformulate the question of existence of tight wavelet frame in terms of the existence of the sum of squares decomposition of a single trigonometric polynomial with real coefficients. Our main result confirms the existence of such decompositions in the two-dimensional case. We also give sufficient conditions for existence of tight wavelet frames in the dimension d ≥ 3 and illustrate our results with several examples.

Acta Applicandae Mathematicae, 2009

An s-elementary normalized tight frame wavelet (associated with an expansive matrix A as its dilation matrix) is a normalized tight frame wavelet whose Fourier transform is of the form 1 √ 2π