A characterization of wavelet sets on Vilenkin groups with its application to construction of MRA wavelets

It is well known that generalized Walsh functions can be considered as characters of Vilenkin groups (see, e.g., [6], [11]). For wavelets on Vilenkin groups most of the results relate to the locally compact group G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> , which is defined by a fixed integer p ≥ 2 (the case p = 2 corresponds to the Cantor group). In this paper, we are interested in an nonstationary MRA and related wavelets for the space L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ); for the stationary case see [2], [4], [5], and references therein. Furthermore, we give an algorithm for construction of nonstationary wavelets on the Vilenkin group associated with a sequence {p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> } <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> , p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> ≥ 2, of natural numbers.

We present two ways to construct frames on the locally compact Cantor dyadic group. The first approach gives a Parseval frame related to the generalized Walsh–Dirichlet kernel, while the second approach includes the Daubechies-type “admissible condition” and leads to dyadic compactly supported wavelet frames. The corresponding wavelet constructions on the Cantor and Vilenkin groups (as well as on the half-line \( {\mathbb{R}_{+} } \)) requires an additional constraint related to the requirement that the masks have no blocking sets.

In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.

Wavelet sets provide a range of wavelets that can be analyzed with ease. The constructions of wavelet sets and scaling sets in different settings are studied in several papers in the literature. Wavelet sets provide a good source of examples and counterexamples in wavelet theory. In this paper we have studied properties of (multi)wavelet sets and associated wavelets for Vilenkin group. Further, results related to scaling sets and generalized scaling sets are given along with some characterizations.

Multiresolution analysis (MRA) is considered as the heart of wavelet theory. The concept of MRA provides an elegant tool for the construction of wavelets. An MRA is an increasing family of closed subspaces \(\left \{V _{j}: j \in \mathbb{Z}\right \}\) of \(L^{2}(\mathbb{R})\) such that \(\bigcap _{j\in \mathbb{Z}}V _{j} = \left \{0\right \},\,\bigcup _{j\in \mathbb{Z}}V _{j}\) is dense in \(L^{2}(\mathbb{R})\) and which satisfies f ∈ V j if and only if f(2⋅ ) ∈ V j+1. Furthermore, there exists an element ϕ ∈ V 0 such that the collection of integer translates of function \(\upphi,\,\left \{\upphi (\cdot - k): k \in \mathbb{Z}\right \}\) represents a complete orthonormal system for V 0. The function ϕ is called the scaling function or the father wavelet. This classic concept of MRA has been extended in various ways in recent years. These concepts are generalized to \(L^{2}(\mathbb{R}^{d})\), to lattices different from \(\mathbb{Z}^{d}\), allowing the subspaces of MRA to be generated by Riesz basis instead of orthonormal basis, admitting a finite number of scaling functions, replacing the dilation factor 2 by an integer M ≥ 2 or by an expansive matrix \(A \in GL_{d}(\mathbb{R})\) as long as \(A \subset A\mathbb{Z}^{d}\). From the last decade, this elegant tool for the construction of wavelet bases have been extensively studied by several authors on the various spaces, namely, abstract Hilbert spaces, locally compact Abelian groups, Cantor dyadic groups, Vilenkin groups, local fields of positive characteristic, p-adic fields, Hyrer-groups, Lie groups, zero-dimensional groups. Notice that the technique is similar to that in the real case of \(\mathbb{R}\) while the mathematical treatment needs ones conscientiousness.

On Wavelets and Prewavelets with Vanishing Moments in Higher Dimensions

Construction of MRA and non-MRA wavelet sets on Cantor dyadic group

MRA wavelets have been widely studied in recent years due to their applications in signal processing. In order to understand the properties of the various MRA wavelets, it makes sense to study the topological structure of the set of all MRA wavelets. In fact, it has been shown that the set of all MRA wavelets (in any given dimension with a fixed expansive dilation matrix) is path-connected. The current paper concerns a class of functions more general than the MRA wavelets, namely normalized tight frame wavelets with a frame MRA structure. More specifically, it focuses on the parallel question on the topology of the set of all such functions (in the given dimension with a fixed dilation matrix): is this set path-connected? While we are unable to settle this general path-connectivity problem for the set of all frame MRA normalized tight frame wavelets, we show that this holds for a subset of it. An s-elementary frame MRA normalized tight frame wavelets (associated with a given expansive matrix A as its dilation matrix) is a normalized tight frame wavelet whose Fourier transform is of the form \(\frac{1}{\sqrt{2\pi}}\chi_{E}\) for some measurable set E⊂ℝd. In this paper, we show that for any given d×d expansive matrix A, the set of all (A-dilation) s-elementary normalized tight frame wavelets with a frame MRA structure is also path-connected.

In this paper, we are concerned with orthonormal wavelets with dilation factor [Formula: see text]. In particular, we shall restrict ourselves to band-limited wavelets and present MRA wavelets and non-MRA wavelets with dilation factor [Formula: see text] for better frequency localization by proposing several types of constructions.

A construction for providing single dyadic orthonormal wavelets in Euclidean space ℝd is given. It is called the general neighborhood mapping construction. The fact that one wavelet is sufficient to generate an orthonormal basis for L2(ℝd) is the critical issue. The validity of the construction is proved, and the construction is implemented computationally to provide a host of examples illustrating various geometrical properties of such wavelets in the spectral domain. Because of the inherent complexity of these single orthonormal wavelets, the method is applied to the construction of single dyadic tight frame wavelets, and these tight frame wavelets can be surprisingly simple in nature. The structure of the spectral domains of the wavelets arising from the general neighborhood mapping construction raises a basic geometrical question. There is also the question of whether or not the general neighborhood mapping construction gives rise to all single dyadic orthonormal wavelets. Results are proved giving partial answers to both of these questions.

Gabardo and Yu first considered using integral self-affine tiles in the Fourier domain to construct wavelet sets and they produced a class of compact wavelet sets with certain self-similarity properties. In this paper, we generalize their results to the integral self-affine multi-tiles setting. We characterize some analytic properties of integral self-affine multi-tiles under certain conditions. We also consider the problem of constructing (multi)wavelet sets using integral self-affine multi-tiles.

Osiris Wavelets in Three Dimensions

Let A be any 2×2 real expansive matrix. For any A-dilation wavelet ψ, let \(\widehat{\psi}\) be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of \((f\widehat{\psi})\) 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 L2(ℝ2) norm topology for any such matrix A.

We describe three types of compactly supported wavelet frames associated with Walsh functions: (1) MRA-based tight frames, (2) frames obtained from the Daubechies-type “admissible condition”, and (3) frames based on the Walsh–Parseval type kernels. Parametric wavelet sets for Vilenkin groups and some related results are also discussed.

We present a collection of easily stated open problems in wavelet theory and we survey the current status of answering them. This includes a problem of Larson ((2007) Unitary systems and Wavelet sets. In: Wavelet analysis and applications. Applied and Numerical Harmonic Analysis. Birkhauser, Basel, pp 143–171) on minimally supported frequency wavelets. We show that it has an affirmative answer for MRA wavelets.