矩阵值函数的多项式插值与高斯求积

基于有理插值的求积公式

Quadrature formulas based on rational interpolation

本文研究基于有理插值的求积公式，涵盖高斯与正交两类，证明其涉及变权重正交多项式，并给出节点渐近分布、交错与单调性结果。

长球面波函数求积与插值公式的详细分析

Detailed analysis of prolate quadratures and interpolation formulas

本文详细分析了长球面波函数（PSWF）的性质，并设计了基于PSWF的求积公式和插值公式，实现了O(n+c)或O(c)复杂度的数值算法，显著提高了大带宽限制c下的计算效率。

经典正交多项式根或极值点多项式插值的显式重心权重

Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials

本文通过建立重心权重与高斯求积之间的联系，利用正交多项式的降算子，推导出包括Gauss-Radau和Gauss-Lobatto点在内的所有已知及新的重心权重显式公式，并基于Hale-Townsend的O(n)算法实现了O(n)计算方案。

Bitangential interpolation in generalized Schur classes

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in a previous paper.

de Branges-Rovnyak spaces and norm-constraint interpolation

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$. A companion survey provides equivalent definitions and basic properties of these spaces as well as applications to function theory and operator theory. The present survey brings to the fore more recent applications to a variety of more elaborate function theory problems, including $H^\infty$-norm constrained interpolation, connections with the Potapov method of Fundamental Matrix Inequalities, parametrization for the s…

Pointwise convergence of Walsh--Fourier series of vector-valued functions

We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p< \infty$, and a UMD space $Y$, the Walsh-Fourier series of $f \in L ^{p}(0,1;Y)$ converges pointwise, provided that $Y$ is a complex interpolation space $Y=[X,H]_θ$ between another UMD space $X$ and a Hilbert space $H$, for some $θ\in(0,1)$. Apparently, all known examples of UMD spaces satisfy this condition.