关于a.e.系数的完全刻划。收敛正交级数

A unifying perspective on linear continuum equations prevalent in science. Part VI: rapidly converging series expansions for their solution

We obtain rapidly convergent series expansions of resolvents of operators taking the form ${\bf A}=Γ_1{\bf B}Γ_1$ where $Γ_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an operator that acts locally in real space. Such resolvents arise naturally when one wants to solve any of the large class of linear physical equations surveyed in Parts I, II, III, and IV that can be reformulated as problems in the extended abstract theory of composites. We show how the information about the spectrum of ${\bf A}$ can be used to greatly improve the convergence rate.

Converging expansions for Lipschitz self-similar perforations of a plane sector

In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the " functional analytic approach " of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $ε$ for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $ε$ tends to zero is described not only by asymptotic series in powers of $ε$, but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator wh…

基于收敛广义幂级数的扰动锥形区域上的Dirichlet问题

Dirichlet problem on perturbed conical domains via converging generalized power series

针对收敛到带锥形奇异性区域的小参数摄动域上的Dirichlet问题，提出基于Laplace-Beltrami本征函数的双尺度交叉截断方法，证明解的分数幂级数在H¹中正规收敛，放宽了泛函分析方法的正则性要求。

关于正交多项式的几点注记

A few remarks on orthogonal polynomials

已知一个给定的、无限支撑的分布的矩序列，我们很快得到：关于该分布正交的一元多项式$\left\{ p_{n}\right\} _{n\geq 0}$的幂级数展开的系数，$\left\{ p_{n}\right\} _{n\geq 0}$在$p_{j},$ $j\leq n$级数中的展开系数，$\left\{ p_{n}\right\} _{n\geq 0}$族的3项递归的系数的两个序列，- 所谓的“线性化系数”，即$p_{j},$ $j\leq m+n.$ \newline系列中的$% p_{n}p_{m}$的展开系数。相反，假设已知给定正交多项式$\left\{ p_{n}\right\} _{n\geq 0},$族的3项递归的系数的两个序列，我们借助它们表示：$p_{n}$的幂级数展开的系数，$x^{n}$在$p_{j},$的级数中的展开的系数$j\leq n,$矩的分布使得多项式$\left\{ p_{n}\right\} _{n\geq 0}$正交。\newline进一步有两个不同的正交多项式族$\left\{ p_{n}\right\} _{n\geq 0}$和$\left\{ q_{n}\right\} _{n\geq 0}$，并知道它们中的每一个的3项递归序列，我们给出这两个多项式族之间的所谓“连接系数”序列。这是$q_{j},$ $j\leq n.$ \newlin…

单侧正交多项式与$SU(2)$值非线性傅里叶级数的一个逐点收敛结果

One sided orthogonal polynomials and a pointwise convergence result for $SU(2)$-valued nonlinear Fourier series

本文研究$SU(2)$值非线性傅里叶级数与单位圆上正交多项式的关系，证明再生核的普适性收敛结果，并建立多项式乘积逐点收敛与零点及局部参数的联系，最后在特定条件下证明沿缺项序列的几乎处处收敛。