Stefan Bildea, Dorin Ervin Dutkay, Gabriel Picioroaga
We construct a multiresolution theory for spaces bigger then L^2(R). For a good choice of the dilation and translation operators on these larger spaces, it is possible to build singly generated wavelet bases, thus obtaining examples of "super-wavelets"
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We provide a characterization of wavelets on local fields of positive characteristic based on results on affine and quasi affine frames. This result generalizes the characterization of wavelets on Euclidean spaces by means of two basic equations. We also give another characterization of wavelets. Further, all wavelets which are associated with a multiresolution analysis on a such a local field are also characterized.
The concept of $p$-adic quincunx Haar MRA was introduced and studied in~\cite{KS10}. In contrast to the real setting, infinitely many different wavelet bases are generated by a $p$-adic MRA. We give an explicit description for all wavelet functions corresponding to the quincunx Haar MRA. Each one generates an orthogonal basis, one of them was presented in~\cite{KS10}. A connection between quincunx Haar bases and two-dimensional separable Haar MRA is also found.
We give a characterization of a class of band-limited wavelets of $L^2({\mathbb R})$ and show that none of these wavelets come from a multiresolution analysis (MRA). For each $n\geq 2$, we construct a subset $S_n$ of ${\mathbb R}$ which is symmetric with respect to the origin. We give necessary and sufficient conditions on a function $ψ\in L^2({\mathbb R})$ with supp $\hatψ\subseteq S_n$ to be an orthonormal wavelet. This result generalizes the characterization of a class of wavelets of E. Hernández and G. Weiss. The dimension functions associated with these wavelets are also computed explicit…
For an arbitrary matrix dilation, any integer n and any integer/semi-integer c, we describe all masks that are symmetric with respect to the point c and have sum rule of order n. For each such mask, we give explicit formulas for wavelet functions that are point symmetric/antisymmetric and generate frame-like wavelet system providing approximation order n. For any matrix dilations (which are appropriate for axial symmetry group on R^2 in some natural sense) and given integer n, axial symmetric/antisymmetric frame-like wavelet systems providing approximation order n are constructed. Also, for se…
A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\bR^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the asso…
For Vilenkin group only the existence of multiwavelets associated with multiresolution analysis (MRA) is known. In this paper, we have shown that by using wavelet sets we can also construct single wavelet in case of Vilenkin group which are not associated with MRA.We have given characterization of single and multi-wavelet sets on Vilenkin group. Further, we have studied generalized scaling sets, some of their properties and relation between wavelet sets and generalized scaling sets.
A characterization of wavelet sets on Vilenkin groups with its application to construction of MRA wavelets
本文在Vilenkin群上建立小波集的特征刻画，为小波构造提供新方法，并阐明多分辨分析与小波集确定的小波之间的关系，正面回答了Mahapatra和Singh提出的一个问题。
We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating a MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate Haar MRA. We also suggest a method of constructing sets of wavelet functions and prove that any set of wavelet functions generates a $p$-adic wavelet frame.
Multiresolution Analysis for Compactly Supported Interpolating Tensor Product Wavelets (long version)
本研究运用巴拿赫空间张量积理论，为两类n维函数空间构建插值张量积MRA及其对偶MRA，并将Besov空间范数等价结果推广到n维情形。
We consider the construction of orthonormal directional wavelet bases in the multi-resolution analysis (MRA) framework with quincunx dilation downsampling. We show that the Parseval frame property in MRA is equivalent to the identity summation and shift cancellation conditions on M functions, which essentially characterize the scaling (father) function and all directional (mother) wavelets. Based on these two conditions, we further derive sufficient conditions for orthonormal bases and build a family of quasishearlet orthonormal bases, that has the same frequency support as that of the least r…
This paper is devoted to wavelet analysis on adele ring $\bA$ and the theory of pseudo-differential operators. We develop the technique which gives the possibility to generalize finite-dimensional results of wavelet analysis to the case of adeles $\bA$ by using infinite tensor products of Hilbert spaces. The adele ring is roughly speaking a subring of the direct product of all possible ($p$-adic and Archimedean) completions $\bQ_p$ of the field of rational numbers $\bQ$ with some conditions at infinity. Using our technique, we prove that $L^2(\bA)=\otimes_{e,p\in\{\infty,2,3,5,...}}L^2({\bQ}_{…
Special-Affine Wavelets: Multi-Resolution Analysis and Function Approximation in L^2(R)
本文提出了一种基于特殊仿射傅里叶变换的多分辨率分析方法，用于在 $ L^2(\mathbb R) $ 中构建正交基组并实现函数逼近。
Approximation by quasi-projection operators and dual wavelet frames in Sobolev spaces
研究Sobolev空间中拟投影算子为函数及其导数提供同时逼近和密度阶的条件，利用所得准则赋予对偶Sobolev空间中基于MRA的对偶小波框架所需同时逼近性质。
The concept of super-wavelet was introduced by Balan, and Han and Larson over the field of real numbers which has many applications not only in engineering branches but also in different areas of mathematics. To develop this notion on local fields having positive characteristic we obtain characterizations of super-wavelets of finite length as well as Parseval frame multiwavelet sets of finite order in this setup. Using the group theoretical approach based on coset representatives, further we establish Shannon type multiwavelet in this perspective while providing examples of Parseval frame (mul…
We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space $L^2(\br)\oplus...\oplus L^2(\br)$.
We define an affine structure on $\ltwor\oplus...\oplus\ltwor$ and, following some ideas developed in \cite{Dut1}, we construct a local trace function for this situation. This trace function is a complete invariant for a shift invariant subspace and it has a variety of properties which make it easily computable. The local trace is then used to give a characterization of super-wavelets and to analyze their multiplicity function, dimension function and spectral function. The "$n\times$" oversampling result of Chui and Shi \cite{CS} is refined to produce super-wavelets.
Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space $\H$ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function $m$ which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space $\H$ is $L^2(\mathbb R^n)$, the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function $m$ satisfying a consistency condi…
In this paper, we study a newly developed shearlet system on bounded domains which yields frames for $H^s(Ω)$ for some $s\in \mathbb{N}$, $Ω\subset \mathbb{R}^2$. We will derive approximation rates with respect to $H^s(Ω)$ norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-based algorithm for the solution of an e…
Motivated by potential applications in multiplexing and by recent results on Gabor analysis with Hermite windows due to Gröchenig and Lyubarskii, we investigate vector-valued wavelet transforms and vector-valued wavelet frames, which constitute special cases of super-wavelets, with a particular attention to the case when the analyzing wavelet vector is related to Fourier transforms of Laguerre functions. We construct an isometric isomorphism between $L^{2}(\mathbb{R}^{+},\mathbf{C}^{n})$ and poly-Bergman spaces, with a view to relate the sampling sequences in the poly-Bergman spaces to the wav…
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 $π(\G)ψ$, where $π$ is a unitary representation of a wavelet group and $\G$ is the abstract pseudo-lattice $\G$. We prove a condition in order that a Parseval frame $π(\G)ψ$ can be dilated to an orthonormal basis of the form $τ(\G)Ψ$ 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 fam…
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