Mexican hat wavelet transform of distributions

Theory of Weierstrass transform is exploited to derive many interesting new properties of the Mexican hat wavelet transform. A real inversion formula in the differential operator form for the Mexican hat wavelet transform is established. Mexican hat wavelet transform of distributions is defined and its properties are studied. An approximation property of the distributional wavelet transform is investigated which is supported by a nice example.     

Horospherical transform on Riemannian symmetric manifolds of noncompact type

We discuss I. M. Gelfand’s project of rebuilding the representation theory of semisimple Lie groups on the basis of integral geometry. The basic examples are related to harmonic analysis and the horospherical transform on symmetric manifolds. Specifically, we consider the inversion of this transform on Riemannian symmetric manifolds of noncompact type. In the known explicit inversion formulas, the nonlocal part essentially depends on the type of the root system. We suggest a universal modification of this operator.     

Moment-recovered approximations of multivariate distributions: The Laplace transform inversion

The moment-recovered approximations of multivariate distributions are suggested. This method is natural in certain incomplete models where moments of the underlying distribution can be estimated from a sample of observed distribution. This approach is applicable in situations where other methods cannot be used, e.g. in situations where only moments of the target distribution are available. Some properties of the proposed constructions are derived. In particular, procedures of recovering two types of convolutions, the copula and copula density functions, as well as the conditional density function, are suggested. Finally, the approximation of the inverse Laplace transform is obtained. The performance of moment-recovered construction is illustrated via graphs of a simple density function.     

Inverse radon transform with one-dimensional wavelet transform

1. IntroductionSince Radon obtained the inverse formula of Radon transform in 1917, different inversemethods such as Fourier inversion, convolution back-projection inversion etc. have beeninvestigated['l']. Wavelet as a useful tool is interested in the inversion of Radon transformin recent years['--']. The application of wavelet analysis to Radon transform was proposedin I4] and [5]. An inversion formula based on continuous wavelet transform was derivedin [6] and [7]. This formula was based…     

关于Camassa-Holm方程动量密度的紧支集在半轴上的估计

主要考虑在半轴上Camassa-Holm方程解的动量密度紧支集大小的估计,方法是根据区间长度与区间特征值的关系,通过估计第一Dirichlet特征值来估计动量密度紧支集的长度.因为知道动量密度紧支集外解的性态,所以通过估计动量密度支集的大小可以得到方程解的更多信息.     

The FBI transform on compact manifolds

We present a geometric theory of the Fourier-Bros-Iagolnitzer transform on a compact manifold . The FBI transform is a generalization of the classical notion of the wave-packet transform. We discuss the mapping properties of the FBI transform and its relationship to the calculus of pseudodifferential operators on . We also describe the microlocal properties of its range in terms of the ``scattering calculus' of pseudodifferential operators on the noncompact manifold .

     

An inversion formula for the dual horocyclic Radon transform on the hyperbolic plane

Consider the Poincare unit disk model for the hyperbolic plane H ². Let Ξ be the set of all horocycles in H ² parametrized by (θ, p), where e^iθ is the point where a horocycle ξ is tangent to the boundary |z| = 1, and p is the hyperbolic distance from ξ to the origin. In this paper we invert the dual Radon transform R* : μ(θ, p) → (z) under the assumption of exponential decay of μ and some of its derivatives. The additional assumption is that P_m(d/dp)(μ_m(p)e^p) be even for all m ∈ ?. Here P_m(d/dp) is a family of differential operators introduced by Helgason, and μ_m(p) are the coefficients of the Fourier series expansion of μ(θ, p). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mehler-Fock transforms of generalized functions via the method of adjoints

In this paper we analyse the Mehler-Fock transform of generalized functions via the method of adjoints. For a distribution of compact support, we prove that its Mehler-Fock transform agrees with its transform via the kernel method. A Paley-Wiener type theorem is established.

     

The Lambert Function on the Solution of a Delay Differential Equation

Recently, a new method for computing the analytical solution of a delay differential equation was developed considering a constant initial function. It is based on the existence of a specific class of polynomials in the delay. In this article, we extend this new method to the case of a continuous initial function. We also show the relationship between the new solution's method and the solution expressed in terms of the Lambert function.     

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.     

关于广义Aluthge变换的数值域

设T是作用在希尔伯特空间H上的有界线性算子,本文研究T的广义Aluthge变换和广义＊-Aluthge变换,并且得到T的广义Aluthge变换的数值域和广义＊-Aluthge变换的数值域相等．     

A study on the convergence conditions of generalized differential transform method

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.     

The Wiener transform on the Besicovitch spaces

In his fundamental research on generalized harmonic analysis, Wiener proved that the integrated Fourier transform defined by is an isometry from a nonlinear space of functions of bounded average quadratic power into a nonlinear space of functions of bounded quadratic variation. We consider this Wiener transform on the larger, linear, Besicovitch spaces defined by the norm . We prove that maps continuously into the homogeneous Besov space for and , and is a topological isomorphism when .

     

Radon变换反演问题的数值方法

利用Tchebycheff多项式和古典Radon变换反演公式,本文得到了Tchebycheff变换对,从而导出了数值反演结果．     

The formation of compact groups of galaxies

In the compact group of galaxies the galaxies can merge into a few massive ones in a very short time, so they must be formed very recently. On the other hand, according to the theory of structure formation, the denser system should form earlier. By analyzing the apparent paradox, we suggest that the merging process of CDM halo plays an important role in the formation of the compact groups of galaxies: it delays the formation of compact groups of galaxies, and makes the pups of galaxies much denser. Project partially supporter1 hy the Ministry of Science and Technology (of China).     

The compact support principle for differential inequalities with gradient terms

We give some sufficient conditions to guarantee the validity of the Compact Support Principle for solutions of some singular elliptic differential inequalities on complete, non-compact manifolds. Our results apply in particular to the p-Laplace and the Mean Curvature operators. We also show that our results are sharp in many important cases.     

Kurzweil-Henstock type integral on zero-dimensional group and some of its applications

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case. Supported by RFFI-08-01-00669.