N. K. Smolentsev, P. N. Podkur
In this paper it is shown, that B-splines are scaling functions for any natural N. For any natural N construction of Haar wavelets, Kotelnikov-Shannon wavelets, nonorthogonal wavelets based on B-splines is given. Examples of construction of filters of N-channel decomposition of signal and filters of reconstruction based on B-splines are given
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In this paper, Meyer wavelets with an arbitrary integer scaling factor $N>2$ are defined using wavelets with multiple scaling factors $MN>2$. Expressions for frequency functions of wavelets and corresponding filters are obtained.
In this paper a simple method of construction of scaling function $ϕ(x)$ and orthogonal wavelets with the compact support for any natural coefficient of scaling $N\ge 2$ is given. Examples of construction of wavelets for coefficients of scaling N=2 and N=3 are produced.
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.
In the present work we provide the bounds for Daubechies orthonormal wavelet coefficients for function spaces $\mathcal{A}_k^p:=\{f: \|(i ω)^k\hat{f}(ω)\|_p< \infty\}$, $k\in\mathbf{N}\cup\{0\}$, $p\in(1,\infty)$.
In this paper, we investigate the wavelet coefficients for function spaces $\mathcal{A}_k^p=\{f: \|(i ω)^k\hat{f}(ω)\|_p\leq 1\}, k\in N, p\in(1,\infty)$ using an important quantity $C_{k,p}(ψ)$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines, which induces a lower bound for the quantity $C_{k,p}(ψ)$ with $ψ$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogona…
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…
Let $Ψ_m^D$ be orthogonal Daubechies wavelets that have m zero moments and let $$ W_{2,p}^k=\{f \in L_2(R):\|(I ω)^k\hat f(ω)\|_p\leq 1\}, \, k \in N. $$ We prove that $$ \lim_{m \to \infty}\, \sup\left\{\frac{|(Ψ_m^D)|}{\|(\hat Ψ_m^D)\|_q}: f \in W_{2, p'}^k\right\}=\frac{\frac{(2π)^{1/p-1/2}}{π^k}\left(\frac{1-2^{1-pk}}{pk-1}\right)^{1/p}}{(2π)^{1/q-1/2}}. $$
This note complements the paper "The quest for optimal sampling: Computationally efficient, structure-exploiting measurements for compressed sensing" [2]. Its purpose is to present a proof of a result stated therein concerning the recovery via compressed sensing of a signal that has structured sparsity in a Haar wavelet basis when sampled using a multilevel-subsampled discrete Fourier transform. In doing so, it provides a simple exposition of the proof in the case of Haar wavelets and discrete Fourier samples of more general result recently provided in the paper "Breaking the coherence barrier…
Two scaling functions $φ_A$ and $φ_B$ for Parseval frame wavelets are algebraically isomorphic, $φ_A \simeq φ_B$, if they have matching solutions to their (reduced) isomorphic systems of equations. Let $A$ and $B$ be $d\times d$ and $s\times s$ \thematrix matrices with $d, s\geq 1$ respectively and let $φ_A$ be a scaling function associated with matrix $A$ and generated by a finite solution. There always exists a scaling function $φ_B$ associated with matrix $B$ such that \begin{equation*} φ_B \simeq φ_A. \end{equation*} An example shows that the assumption on the finiteness of the solutions c…
We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on ${\rm L}^2(\mathbb{R}^d)$. We first prove that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions densely contained in (most of) the coorbit spaces. We also present an examp…
In this paper we present a general approach to multivariate periodic wavelets generated by scaling functions of de la Vallée Poussin type. These scaling functions and their corresponding wavelets are determined by their Fourier coefficients, which are sample values of a function, that can be chosen arbitrarily smooth, even with different smoothness in each direction. This construction generalizes the one-dimensional de la Vallée Poussin means to the multivariate case and enables the construction of wavelet systems, where the set of dilation matrices for the two-scale relation of two spaces of …
Wavelet families arise by scaling and translations of a prototype function, called the {\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a {\em{diction…
In this article, we present a constructive method for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and group theory.
Using the Daubechies conditions of compact support, orthogonal, and regularity, we were able to derive bivariate scaling functions with which to reproduce linear functions (planes). We describe how to create all possible masks of refinement coefficients that satisfy those conditions.
Symmetry, Scaling, and Optimal Time-Frequency Concentration: Minimising the Heisenberg Uncertainty in Piecewise-Polynomial and Wavelet Dictionaries
本文提出函数类层级与定制不确定性算子，证明不确定性在尺度平移下不变，帐篷函数为两个小波字典中时频不确定性的唯一极小化器（U=3/10），且rect的p重自卷积不确定性随p增大单调趋向海森堡下界1/4，为自适应小波设计提供原则性方法。
p-Adic Scaling Set and Generalized Scaling Set
本文旨在 L2(Qp) 中发展 MRA 与小波理论，探讨了尺度集的必要条件，并通过实例刻画了广义尺度集。
A generalization of Mallat's classic theory of multiresolution analysis based on the theory of spectral pairs was considered by Gabardo and Nashed (J. Funct. Anal. 158, 209-241, 1998). In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum $Λ=\left\{0, r/N\right\}+2\mathbb Z$, where $N\ge 1$ is an integer and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear …
Explicit formulae are given for a type of Battle-Lemarié scaling functions and related wavelets. Compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to Nikolskii-Besov spaces on R.
Unique wavelet sign retrieval from samples without bandlimiting
本文证明实值信号可在无带限约束下，通过由Poisson小波及其Hilbert变换线性组合构成的三个小波框架系数幅值实现全局符号意义下的唯一恢复，而复值信号则无法做到。
We show the relevance of a multifractal-type analysis for pointwise convergence and divergence properties of wavelet series: Depending on the sequence space which the wavelet coefficients sequence belongs to, we obtain deterministic upper bounds for the Hausdorff dimensions of the sets of points where a given rate of divergence occurs, and we show that these bounds are generically optimal, according to several notions of genericity.
We use Lorentz polynomials to present the solutions explicitly of equations (6.1.7) of [I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992] and (4.9) of [I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988), no. 7, 909--996] sot that we give an efficient way to prove Daubechies' results on the existence of spline type orthogonal scaling functions and to evaluate Daubechies scaling functions.
This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters $p \in (0,\infty)$, $q \in (0,\infty]$ and $α\in \mathbb{R}$. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated by translations and anisotropic dilations of a single function, where neither the translation no…
In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions $\{ψ_λ\}_{λ\in Λ}\subset L^2(\mathbb{R}^d)$ that constitutes a semi-discrete frame, we ask whether any real-valued function $f \in L^2(\mathbb{R}^d)$ can be uniquely recovered from its unsigned convolutions ${\{|f \ast ψ_λ|\}_{λ\in Λ}}$. We find that under some mild assumptions on the semi-discrete frame and if $f$ has exponential decay at $\infty$, it suffices to know $|f \ast ψ_λ|$ on suitably fine lattices to uniquely determine $f$ (up to a global sign factor). We fur…
Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces $\dot{\mathbf{F}}^α_{p,q}$ for the endpoint case of $p = \infty$ and the full scale of parameters $α\in \mathbb{R}$ and $q \in (0,\infty]$. In particular, a Peetre-type characterization of the anisotropic Besov space $\dot{\mathbf{B}}^α_{\infty,\infty} = \dot{\mathbf{F}}^α_{\infty,\infty}$ is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $\dot{\mathbf{F}}^α_{\infty,q}$.
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.
The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto wavelet subspaces corresponding to the multidimensional multiresolution analysis generated as tensor product of smooth finite scaling functions of one variable.
In this paper, we design mother wavelets for the 1D continuous wavelet transform with some optimality properties. An optimal mother wavelet here is one that has an ambiguity function with minimal spread in the continuous coefficient space (also called phase space). Since the ambiguity function is the reproducing kernel of the coefficient space, optimal windows lead to phase space representations which are "optimally sharp." Namely, the wavelet coefficients have minimal correlations with each other. Such a construction also promotes sparsity in phase space. The spread of the ambiguity function …
The aim of this paper is to show the usefulness of Meyer wavelets for the classical problem of density estimation and for density deconvolution from noisy observations. By using such wavelets, the computation of the empirical wavelet coefficients relies on the fast Fourier transform of the data and on the fact that Meyer wavelets are band-limited functions. This makes such estimators very simple to compute and this avoids the problem of evaluating wavelets at non-dyadic points which is the main drawback of classical wavelet-based density estimators. Our approach is based on term-by-term thresh…
Coorbit Theory for Coefficients in Weighted Lebesgue Spaces, and its Application to the Wavelet Transform and the Short-Time Fourier Transform
本文基于局部紧群可积酉表示构造语音变换取值于加权 Lebesgue 空间的 coorbit 空间，并给出 Banach 框架离散化及小波、Gabor 窗口的适用充分条件。
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