k解析函数的Fourier级数

讨论了复平面上k解析函数的性质,并利用k解析函数的泰勒展开定理研究了k解析函数的Fourier级数,推广了经典的解析函数的Fourier级数理论.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

关于广义Fourier级数的若干结论及其应用

讨论Hilbert空间广义Fourier级数收敛的充分和必要条件,并将相关结果应用于数学分析中具体的Fourier级数上.     

窗口Fourier变换反演公式的级数表示——献给陆善镇教授75华诞

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L^2空间,本文给出重构级数收敛的充分必要条件.     

半空间中一类推广的Poisson积分的增长性质

本文研究了推广的Poisson积分的增长性问题.利用复平面中经典的Hayman定理及其证明方法,通过修改上半空间中的Poisson积分,获得了上半空间中一类位势在无穷远点的增长性质,推广了Hayman定理在高维空间的结果.     

复值函数的广义复模糊积分北大核心

本文给出了非负复值函数关于模糊复测度的广义复模糊积分的几种等价形式,并讨论了由该积分表示的几种复模糊积分方程有解的充要条件.在此基础上,进一步引进了一般复值函数的广义复模糊积分的概念,给出了该积分的一些基本性质,并在一定条件下证明了单调收敛定理.     

复值函数的广义复模糊积分

本文给出了非负复值函数关于模糊复测度的广义复模糊积分的几种等价形式,并讨论了由该积分表示的几种复模糊积分方程有解的充要条件.在此基础上,进一步引进了一般复值函数的广义复模糊积分的概念,给出了该积分的一些基本性质,并在一定条件下证明了单调收敛定理.     

Fourier级数两个收敛定理的关系及反例

Fourier级数的收敛问题一直是很多数学研究者关注的问题,不同的数学分析教材和高等数学教材对收敛定理的表述各不相同。本文通过实例说明这些收敛定理之间不存在包含关系,并进一步说明这些收敛定理都是判别Fourier级数收敛的充分条件,而不是充要条件。     

复Banach空间上的Rouché定理

本文利用（广义）拓扑度的有关性质给出了一般复Ｂａｎａｃｈ空间上全纯映射的（广义）Ｒｏｕｃｈé定理．特别，在复平面上它以经典的Ｒｏｕｃｈé定理为特例．同时，它也给出了多复变全纯映射的相应定理及一些应用．     

拟可加测度空间上广义Sugeno积分的收敛性

在拟可加测度空间上通过引入拟乘算子重新定义广义Sugeno积分,针对依拟可加测度收敛的函数列,应用诱导算子和拟乘算子的运算性质讨论和分析广义Sugeno积分的收敛性,进而获得了这种广义Sugeno积分的单调增收敛定理.     

Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms

This study concerns some new developments of unit analytic signals with non-linear phase. It includes ladder-shaped filter, generalized Sinc function based on non-linear Fourier atoms, generalized sampling theorem for non-bandlimited signals and the notion of multi-scale spectrum for discrete sequences. We first introduce the ladder-shaped filter and show that the impulse response of its corresponding linear time-shift invariant system is the generalized Sinc function as a product of periodic Poisson kernel and Sinc function. Secondly, we establish a Shannon-type sampling theorem based on generalized Sinc function for this type of non-bandlimited signal. We further prove that this type of signal may be holomorphically extended to strips in the complex plane containing the real axis. Finally, we introduce the notion of multi-scale spectrums for discrete sequences and develop the related fast algorithm.     

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.     

Structure theorems for associated homogeneous distributions based on the line

Associated homogeneous distributions (AHDs) with support in the line R are the distributional generalizations of one‐dimensional power‐log functions. In this paper, we derive a number of practical structure theorems for AHDs based on R and being complex analytic with respect to their degree of homogeneity in some region of the complex plane. Each theorem gives a representation that is designed to have a distinct advantage for calculating either convolution products, multiplication products, generalized derivatives and primitives, Fourier transforms or Hilbert transforms of AHDs. Copyright © 2008 John Wiley & Sons, Ltd.     

Paley-Wiener-Type Theorems for a Class of Integral Transforms

A characterization of weighted L₂(I) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley-Wiener-type theorems for these spaces. Unlike the classical Paley-Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line and on the whole line.     

A quadratic-phase integral operator for sets of generalized integrable functions

In this paper, we aim to discuss the classical theory of the quadratic-phase integral operator on sets of integrable Boehmians. We provide delta sequences and derive convolution theorems by using certain convolution products of weight functions of exponential type. Meanwhile, we make a free use of the delta sequences and the convolution theorem to derive the prerequisite axioms, which essentially establish the Boehmian spaces of the generalized quadratic-phase integral operator. Further, we nominate two continuous embeddings between the integrable set of functions and the integrable set of Boehmians. Furthermore, we introduce the definition and the properties of the generalized quadratic-phase integral operator and obtain an inversion formula in the class of Boehmians.     

一类具一阶奇性核的奇异积分方程

当(?)是复平面C上的光滑封闭曲线,k（z）是在(?)所围成的有界闭区域上连续.在其内部解析的函数时.借助于奇异积分算子的广义逆.讨论了具一阶奇性核的正则型奇异积分方程:　在H类中的求解问题.作为应用,作者给出了当k（z）是一类有理函数时的具体解法,从而统一并推广了 Cauchy核和Hilbert核奇异积分方程的经典结果.     

等周不等式与Bol-Fujiwara定理

本文研究著名的Bol-Fujiwara定理.利用积分几何方法和经典的等周不等式,得到了Bol-Fujiwara定理的一个推广(定理1),以及推广了的Bol-Fujiwara定理的逆定理(定理2).     

Fourier restriction for affine arclength measures in the plane

We obtain an analog, uniform for a large class of curves in the plane, of the Fefferman-Zygmund theorem on restriction of the Fourier transform.