We prove a continuity result for the shearlet transform when restricted to the space of smooth and rapidly decreasing functions with all vanishing moments. We define the dual shearlet transform, called here the shearlet synthesis operator, and we prove its continuity on the space of smooth and rapidly decreasing functions over $\mathbb{R}^2\times\mathbb{R}\times\mathbb{R}^\times$. Then, we use these continuity results to extend the shearlet transform to the space of Lizorkin distributions, and we prove its consistency with the classical definition for test functions.
We develop a distributional framework for the shearlet transform $\mathcal{S}_ψ\colon\mathcal{S}_0(\mathbb{R}^2)\to\mathcal{S}(\mathbb{S})$ and the shearlet synthesis operator $\mathcal{S}^t_ψ\colon\mathcal{S}(\mathbb{S})\to\mathcal{S}_0(\mathbb{R}^2)$, where $\mathcal{S}_0(\mathbb{R}^2)$ is the Lizorkin test function space and $\mathcal{S}(\mathbb{S})$ is the space of highly localized test functions on the standard shearlet group $\mathbb{S}$. These spaces and their duals $\mathcal{S}_0^\prime (\mathbb R^2),\, \mathcal{S}^\prime (\mathbb{S})$ are called Lizorkin type spaces of test functions …
In this paper, we introduce the notion of quaternion shearlet transform- which is an extension of the ordinary shearlet transform. Firstly, we study the fundamental properties of quaternion shearlet transforms and then establish some basic results including Moyal's and inversion formulae. Finally, we derive the associated Heisenberg's uncertainty inequality and the corresponding logarithmic version for quaternion shearlet transforms.
The aim of this article is to formulate some novel uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of the Pitt's inequality for the continuous shearlet transforms, then we formulate the Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner's inequality in the Fourier domain. Secondly, we consider a logarithmic Sobolev inequality for the continuous shearlet transforms which has a dual relation with Beckner's inequality. Thirdly, …
We show that the cone-adapted shearlet coefficients can be computed by means of the limited angle horizontal and vertical (affine) Radon transforms and the one-dimensional wavelet transform. This yields formulas that open new perspectives for the inversion of the Radon transform.
Continuous Shearlet Frames and Resolution of the Wavefront Set
近年来，方向多尺度变换，如曲波或剪切波变换，得到了相当大的关注。其原因是这些变换-与小波等更传统的变换不同-能够有效地处理具有边缘特征的数据。[G. Kutyniok，D.拉贝特Resolution of the Wavefront Set using continuous Shearlets，Trans.AMS 361（2009），2719 - 2754] Kutyniok和Labate证实了剪切波的这一性质，其中表明，对于在紧凑锥形楔形体中具有频率支持的非常特殊的函数$ψ$，回火分布$f$相对于剪切波$ψ$的剪切波系数的衰减率可以解决这一问题。$f$的波前设置。我们证明了同样的结果可以验证下，更弱的假设$ψ$，即拥有足够多的各向异性消失矩。我们还展示了如何从任何这样的函数构建$L^2(\mathbb{R}^2)$的框架。为了证明我们的声明，我们开发了一种新的方法的基础上适应的Radon变换的剪切波结构。
We introduce two shearlet-based Ginzburg--Landau energies, based on the continuous and the discrete shearlet transform. The energies result from replacing the elastic energy term of a classical Ginzburg--Landau energy by the weighted $L^2$-norm of a shearlet transform. The asymptotic behaviour of sequences of these energies is analysed within the framework of $Γ$-convergence and the limit energy is identified. We show that the limit energy of a characteristic function is an anisotropic surface integral over the interfaces of that function. We demonstrate that the anisotropy of the limit energy…
We find a new and simple inversion formula of the Radon transform RT with the only use of the shearlet system and of well-known properties of RT. No intertwining relation of differential operators in Euclidean space and Radon domain is used. As a consequence, an additive noise is not incremented. Since the continuum theory of shearlets has a straight translation to the discrete theory, we find a fast, stable and computable algorithm that recovers a digital image from noisy samples of the Radon transform preserving edges. In the process, we find a more natural and easier-to-construct density-co…
Sobolev wavefront sets and $2$-microlocal spaces play a key role in describing and analyzing the singularities of distributions in microlocal analysis and solutions of partial differential equations. Employing the continuous shearlet transform to Sobolev spaces, in this paper we characterize the microlocal Sobolev wavefront sets, the $2$-microlocal spaces, and local Hölder spaces of distributions/functions. We then establish the connections among Sobolev wavefront sets, $2$-microlocal spaces, and local Hölder spaces through the continuous shearlet transform.
The shearlet transform from applied harmonic analysis is currently the state of the art when analyzing multidimensional signals with anisotropic singularities. Its optimal sparse approximation properties and its faithful digitalization allow shearlets to be applied to different problems from imaging science, such as image denoising, image inpainting, and singularities detection. The shearlet transform has also be successfully utilized, for instance, as a feature extractor. As such it has been shown to be well suited for image preprocessing in combination with data-driven methods such as deep n…
In this papae we introduce and investigate new 2-microlocal spaces associated with Besov type and Triebel-LIzorkin type spaces. We establish characterizations of these function spaces via the phi transform, the atom and molecular decomposition and wavelet decomposition. As applications we consider the boundedness of the Calderon-Zygmund operator and the pseudo-differential operator on these function spaces
In this paper, the author introduce Triebel-Lizorkin spaces with general smoothness. We present the $φ$-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev embeddings. Also, we establish the smooth atomic and molecular decomposition of these function spaces. To do these we need a generalization of some maximal inequality to the case of general weights.
Riesz Transform Characterization and Fefferman-Stein Decomposition of Triebel-Lizorkin Spaces
设$D\in\mathbb{N}$，$q\in[2,\infty)$和$(\mathbb{R}^D,|\cdot|,dx)$是具有$D$维勒贝格测度的欧氏空间.本文通过小波展开定义的辅助函数空间$\mathrm{WE}^{1,\,q}(\mathbb R^D)$，建立了Triebel-Lizorkin空间$\dot{F}^0_{1,\,q}(\mathbb{R}^D)$的里斯变换刻画.由此得到了Triebel-Lizorkin空间$\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$的Festhiman-Stein分解.最后，作者给出了一个显式的例子，说明$\dot{F}^0_{1,\,q}(\mathbb{R}^D)$严格包含在$\mathrm{WE}^{1,\,q}(\mathbb{R}^D)$和，通过对偶，$\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$严格包含在$\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$。虽然当$D=1$由C.- C. Lin等人[Michigan Math.J.62（2013），691-703]，如C. C. Lin等人，在$D=1$情况下使用的方法不能应用于$D\ge2$情况，这需要一些新的技巧。
We introduce almost diagonal matrices in the setting of (anisotropic) discrete homogeneous Triebel-Lizorkin type spaces and homogeneous modulation spaces, and it is shown that the class of almost diagonal matrices is closed under matrix multiplication. We then connect the results to the continuous setting and show that the "change of frame" matrix for a pair of time-frequency frames, with suitable decay properties, is almost diagonal. As an application of this result, we consider a construction of compactly supported frame expansions for homogeneous decomposition spaces of Triebel-Lizorkin typ…
Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and pote…
Fix $λ>0$. Consider the Bessel operator $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ on $\mathbb{R_+}$, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel--Lizorkin spaces associated with $\triangle_λ$. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.
The paper deals with the problem under which conditions for the parameters $s_1,s_2\in\mathbb{R}$, $1\leq p,q_1,q_2\leq\infty$ the Fourier transform $\mathcal{F}$ is a nuclear mapping from $A^{s_1}_{p,q_1}(\mathbb{R}^n)$ into $A^{s_2}_{p,q_2}(\mathbb{R}^n)$, where $A\in\{B,F\}$ stands for a space of Besov or Triebel-Lizorkin type, and $n\in\mathbb{N}$. It extends the recent paper arXiv:2112.04896 where the compactness of $\mathcal{F}$ acting in the same type of spaces was studied.
