连续切波变换反演公式的级数表示

我们主要研究连续切波变换反演公式的级数表示.首先引入两类由切波变换反演公式定义的无穷级数和有限级数,并研究了由Kittipoom等人介绍的切波生成空间,得到这个切波生成空间的一些重要性质.其次利用这些结果显示:对于这个切波生成空间,当采样密度趋于无穷时由我们定义的无穷级数按L~2-范数收敛于重构函数;对于可允许函数空间,当采样密度趋于无穷时由我们定义的有限级数按L~2-范数收敛于重构函数.     

连续切波变换的高维奇异性分析

在高维数据处理过程中,确定高维平方可积函数的奇异性有着重要的意义,它可作为模式识别、数据挖掘、频谱分析、大型机械故障诊断、航空航天、遥感与控制以及三维图像处理的基础.本文首先给出高维平方可积函数的连续切波变换重构公式;其次研究几种特殊函数的切波系数的衰减性质;最后运用重构公式中的切波系数刻画平方可积函数的奇异支撑集.本文的结果推广了Kutyniok和Dahlke等人给出的一些已知结果.     

Bessel变换的一个下界

Bessel逆问题在物理、化学和工程学等诸多领域有重要应用.解决线性逆问题的传统方法不适合处理具有奇异性曲线边缘的二元函数.鉴于切波对这一类函数的最优表示能力,相关文献采用切波方法研究Bessel逆问题,构造了目标函数的切波域值估计器,得到了它在函数空间V中积分均方差收敛阶的上界.在此基础上利用统计理论给出其最小最大风险的一个下界,证明了在估计Bessel逆问题时此估计器是最优的.     

约化Banach 代数的谱不变性和双曲群

主要研究了可数离散群的性质 SRD, 并证明了具有性质 SRD的群上的速降函数全体组成的空间是约化Banach代数 谱不变的稠密子代数. 最后作为例子给出了双曲群具有性质 SRD.     

小波树逼近及其应用

利用“构造性贪婪算法(CGS)”构造目标函数的小波树逼近. 首先定义了一个函数类, 对此函数类中的每个函数, 由CGS生成的分片多项式逼近都具有给定的收敛阶. 其次通过研究所定义函数类的嵌入性质讨论了该函数类和其他已知函数空间的关系. 在小波树逼近领域, 给出了使小波树逼近达到最优收敛阶的一个充分条件. 最后证明, 如果树结构是用CGS生成的, 则相应的小波树逼近具有最优收敛阶.     

局部凸空间中的Widder-Arendt定理与积分半群

发展了Arendt建立的关于Banach空间值函数Laplace变换的著名结果,建立了序列完备局部凸空间框架下的Widder-Arendt定理(定理1.1).引入了该空间中的积分半群并利用定理1.1建立了此类半群的生成定理及一些基本性质.作为具体的例子,给出了关于几类局部凸空间中椭圆微分算子何时为积分半群生成元的结果.     

一类不可分的 Bα函数空间

Ba 函数空间是由丁夏畦教授和罗佩珠教授在文[1—3]中引入的.它已得到了大量应用(参见[4—8]).由于 Ba 函数空间是由一列 L_p 空间产生的(定义见第二节),人们自然会提出如下问题:问题 A.若已知某性质对 L_p 空间成立,是否可得出该性质对 Ba 函数空间也成立     

球面光滑核漂移的Lp逼近

将光滑的球面基函数φ嵌入到由一个不充分光滑的球面基函数ψ生成的本性空间Nψ中,并在Lp度量下研究由φ的变换生成的函数在空间Nψ中的逼近性质,得到了该Lp逼近的误差估计.     

球面光滑核漂移的L~p逼近

将光滑的球面基函数φ嵌入到由一个不充分光滑的球面基函数Ψ生成的本性空间N_Ψ中,并在L~p度量下研究由φ的变换生成的函数在空间N_Ψ中的逼近性质,得到了该L~p逼近的误差估计.     

半直线伸缩调制框架集

本文研究右半直线平方可积函数空间L^2(R+)中的一类伸缩调制系.实际问题中时间变量不可取负值,L^2(R+)可模拟因果信号空间.但因R+按加法不能作成一个群,它不容许小波与Gabor系.我们研究L^2(R+)中由特征函数生成的伸缩调制系(MD-系)框架,引入了R+中MD-框架集的概念,利用"伸缩等价"与"基数函数"方法刻画了L^2(R+)中MD-Bessel集与完备集;得到了关于MD-Riesz基集的两个充分条件,并证明了通过对MD-Riesz基集进行有限可测分解可得到MD-框架集.     

Necessary conditions and sufficient conditions of irregular shearlet frames

In this paper, necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied. We show that the irregular shearlet systems to possess upper frame bounds, the space‐scale‐shear parameters must be relatively separated. We prove that if the irregular shearlet systems possess the lower frame bound and the space‐scale‐shear parameters satisfy certain condition, then the lower shearlet density is strictly positive. We apply these results to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems. We prove that for a feasible class of shearlet generators introduced by P. Kittipoom et al., each relatively separated sequence with sufficiently hight density will generate a frame. Explicit frame bounds are given. We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space‐scale‐shear parameters and the generating function undergo small perturbations. Explicit stability bounds are given. Using pseudo‐spline functions of type I and II, we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results.     

Inversion Formula for Shearlet Transform in Arbitrary Space Dimensions

In this article, we study the convergence of the inverse shearlet transform in arbitrary space dimensions. For every pair of admissible shearlets, we show that although the integral involved in the inversion formula from the continuous shearlet transform is convergent in the L² sense, it is not true in general whenever pointwise convergence is considered. We give some su?cient conditions for the pointwise convergence to hold. Moreover, for any pair of admissible shearlets we show that the Riemannian sums defined by the inverse shearlet transform are convergent to the original function as the sampling density tends to infinity.     

Pseudorandom number generators based on random covers for finite groups

Random covers for finite groups have been introduced in Magliveras et?al. (J Cryptol 15:285–297, 2002), Lempken et?al. (J Cryptol 22:62–74, 2009), and Svaba and van Trung (J Math Cryptol 4:271–315, 2010) for constructing public key cryptosystems. In this article we describe a new approach for constructing pseudorandom number generators using random covers for large finite groups. We focus, in particular, on the class of elementary abelian 2-groups and study the randomness of binary sequences generated from these generators. We successfully carry out an extensive test of the generators by using the NIST Statistical Test Suite and the Diehard battery of tests. Moreover, the article presents argumentation showing that the generators are suitable for cryptographic applications. Finally, we include performance data of the generators and propose a method of using them in practice.     

Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.     

Dualizable Shearlet Frames and Sparse Approximation

Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requires knowledge of a dual frame. However, no closed and easily computable form of any dual frame is known. In this paper, we introduce the class of dualizable shearlet systems, which consist of compactly supported elements and can be proved to form frames for \(L^2({\mathbb {R}}^2)\). For each such dualizable shearlet system, we then provide an explicit construction of an associated dual frame, which can be stated in closed form and is efficiently computed. We also show that dualizable shearlet frames still provide near optimal sparse approximations of anisotropic features.     

Shearlet Smoothness Spaces

The shearlet representation has gained increasingly more prominence in recent years as a flexible and efficient mathematical framework for the analysis of anisotropic phenomena. This is achieved by combining traditional multiscale analysis with a superior ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure.     

Shearlet coorbit spaces and associated Banach frames

In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.     

A class of gap functions for variational inequalities

Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.Corresponding author.