Kwok Hao Lee, Guido L. Weiss
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.
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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…
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。
Multiresolution analysis (MRA) on compact abelian group $G$ has been constructed with epimorphism as a dilation operator. We show a characterization of scaling sequences of an MRA on $L^p(G)$, $1\le p<\infty$. With the help of this scaling sequence we construct a wavelet orthonormal basis of $L^2(G)$.
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.
A generalized filter construction is used to build an example of a non-MRA normalized tight frame wavelet for dilation by 2 in $L^2(\mathbb R)$. This example has the same multiplicity function as the Journé wavelet, yet has a $C^{\infty}$ Fourier transform and can be made to be $C^r$ for any fixed positive integer $r$.
Meyer wavelets for rational dilations
本文证明了任意空间维度下实数膨胀存在平滑带限多分辨率分析及有理膨胀下正交Meyer小波的存在性，并证明良好局部化的正交MRA小波仅存在于有理膨胀中，确立了结果的最优性。
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。
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。
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。
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…
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