Pascal Auscher, Tuomas Hytönen
We bring a precision to our cited work concerning the notion of "Borel measures", as the choice among different existing definitions impacts on the validity of the results.
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Wavelet orthonormal bases for nonexpansive dilations
本文为非扩张膨胀构造不承认小波集且不满足Calderón求和公式的小波正交基，并证明行列式为1的膨胀矩阵存在小波基。
In this article, we follow closely the approach in Hernandez and Weiss's seminal text in describing the construction of an orthonormal wavelet from a multi-resolution analysis. We assume the reader has a modest background in analysis and measure theory.
In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this…
Numerical optimization is used to construct new orthonormal compactly supported wavelets with Sobolev regularity exponent as high as possible among those mother wavelets with a fixed support length and a fixed number of vanishing moments. The increased regularity is obtained by optimizing the locations of the roots the scaling filter has on the interval (pi/2,π). The results improve those obtained by I. Daubechies [Comm. Pure Appl. Math. 41 (1988), 909-996], H. Volkmer [SIAM J. Math. Anal. 26 (1995), 1075-1087], and P. G. Lemarie-Rieusset and E. Zahrouni [Appl. Comput. Harmon. Anal. 5 (1998), …
Discrete Calderon condition for the twisted wavelet system
该数学研究提出离散Calderon条件的类似形式，证明其可作为正交扭曲小波系统在L^{2}(R^{2})中完备的充分条件，丰富了小波分析理论。
On wavelet polynomials and Weyl multipliers
基于小波正交系统，建立新的界，证明log n是几乎处处收敛的韦伊尔乘子，且为最优序列。
Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have Hölder regularity. This is used to build an orthonormal basis of Hölder-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calderón reproducing formula to study and develop function space theory and singular integrals. We discuss the example…
On the Calderón sum formula for wavelet systems
本文证明了在特定条件下，小波系统的Calderón求和公式适用于任意扩张和翻译矩阵，部分解决了Bownik和Lemvig的猜想。
A classical theorem attributed to Naimark states that, given a Parseval frame $\mathcal{B}$ in a Hilbert space $\mathcal{H}$, one can embed $\mathcal{H}$ in a larger Hilbert space $\mathcal{K}$ so that the image of $\mathcal{B}$ is the projection of an orthonormal basis for $\mathcal{K}$. In the present work, we revisit the notion of Parseval frame MRA wavelets from two papers of Paluszyński, Šikić, Weiss, and Xiao (PSWX) and produce an analog of Naimark's theorem for these wavelets at the level of their scaling functions. We aim to make this discussion as self-contained as possible and provid…
We exhibit the necessary range for which functions in the Sobolev spaces $L^s_p$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle-Lemarié wavelets. We also consider the natural extensions to Triebel-Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.
It is rather unexpected, but true, that it is possible to construct reproducing formulae and orthonormal bases of $L^2 (\mathbb{R}^2)$ just by applying the standard one dimensional wavelet action of translations and dilations to the first variable $x_1$ of the generating function $ψ(x_1,x_2)$, $ψ\in L^2 (\mathbb{R}^2)$, i.e., by making use of building blocks $$ψ_{u,s}(x_1,x_2)=s^{-1/2}ψ\left(\frac{x_1-u}{s},x_2\right), \text{where } u\in \mathbb{R}, s>0,$$ in the case of reproducing formulae, and $$ψ_{k,m}(x_1,x_2)=2^{-k/2} ψ\left(\frac{x_1-2^k m}{2^k},x_2 \right), \text{where } k,m\in \mathbb…
Let $({\mathcal X},d,μ)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of $H^1_{\rm at}({\mathcal X})$ and several equivalent characterizations of $H^1_{\rm at}({\mathcal X})$ in terms of wavelets, which are proved useful.
Almost everywhere convergence of a wavelet-type Malmquist-Takenaka series
本文研究了一种波浪型Malmquist-Takenaka级数，证明其在$H^2(\mathbf{T})$中几乎处处收敛。
In any quasi-metric space of homogeneous type, Auscher and Hytönen recently gave a construction of orthonormal wavelets with Hölder-continuity exponent $η>0$. However, even in a metric space, their exponent is in general quite small. In this paper, we show that the Hölder-exponent can be taken arbitrarily close to 1 in a metric space. We do so by revisiting and improving the underlying construction of random dyadic cubes, which also has other applications.
Frames and Bases of Translates of Signals on Undirected Graphs
本文研究图上平移不变空间，推导了信号平移系统构成正交基的条件，并探讨了多生成器构成框架的特性，同时介绍了调制波let系统。
Let $({\mathcal X},d,μ)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, the authors prove that the product $f\times g$ of $f\in H^1_{\rm at}({\mathcal X})$ and $g\in\mathop\mathrm{BMO}({\mathcal X})$, viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ into $L^1…
An inspiration at the origin of wavelet analysis (when Grossmann, Morlet, Meyer and collaborators were interacting and exploring versions of multiscale representations) was provided by the analysis of holomorphic signals, for which the images of the phase of Cauchy wavelets were remarkable in their ability to reveal intricate singularities or dynamic structures, such as instantaneous frequency jumps in musical recordings. Our goal is to follow their seminal work and introduce recent developments in nonlinear analysis. In particular we sketch methods extending conventional Fourier analysis, exp…
Assume that $(X, d, μ)$ is a space of homogeneous type in the sense of Coifman and Weiss. In this article, motivated by the breakthrough work of P. Auscher and T. Hytönen on orthonormal bases of regular wavelets on spaces of homogeneous type, the authors introduce a new kind of approximations of the identity with exponential decay (for short, $\exp$-ATI). Via such an $\exp$-ATI, motivated by another creative idea of Y. Han et al. to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, the authors estab…
In applications, choices of orthonormal bases in Hilbert space H may come about from the simultaneous diagonalization of some specific abelian algebra of operators. It was noticed recently that there is a certain finite set of non-commuting operators F_i, first introduced by engineers in signal processing, which helps to clarify this connection, and at the same time throws light on decomposition possibilities for wavelet packets used in pyramid algorithms. While the operators F_i were originally intended for quadrature mirror filters of signals, recent papers have shown that they are ubiquitou…
It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,μ)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $μ$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend …
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