流形上的多调和及相关核：插值与逼近

The polyharmonic Bergman space for the union of rotated unit balls

In the paper we consider the polyharmonic Bergman space for the union of the rotated unit Euclidean balls. Using so called zonal polyharmonics we derive the formulas for the kernel of this space. Moreover, we study the weighted polyharmonic Bergman space. By the same argument we get the Bergman kernel for this space.

Uniformly bounded Lebesgue constants for scaled cardinal interpolation with Matérn kernels

For $h>0$ and positive integers $m$, $d$, such that $m>d/2$, we study non-stationary interpolation at the points of the scaled grid $h\mathbb{Z}^d$ via the Matérn kernel $Φ_{m,d}$---the fundamental solution of $(1-Δ)^m$ in $\mathbb{R}^d$. We prove that the Lebesgue constants of the corresponding interpolation operators are uniformly bounded as $h\to0$ and deduce the convergence rate $O(h^{2m})$ for the scaled interpolation scheme. We also provide convergence results for approximation with Matérn and related compactly supported polyharmonic kernels.

On the density of polyharmonic splines

This article treats the question of fundamentality of the translates of a polyharmonic spline kernel (also known as a surface spline) in the space of continuous functions on a compact set $Ω\subset \RR^d$ when the translates are restricted to $Ω$. Fundamentality is not hard to demonstrate when a low degree polynomial may be added or when translates are permitted to lie outside of $Ω$; the challenge of this problem stems from the presence of the boundary, for which all successful approximation schemes require an added polynomial. When $Ω$ is the unit ball, we demonstrate that translates of poly…

Wiener-Hopf difference equations and semi-cardinal interpolation with integrable convolution kernels

Let $H\subset\mathbb{Z}^d$ be a half-space lattice, defined either relative to a fixed coordinate (e.g.\ $H = \mathbb{Z}^{d-1}\!\times\!\mathbb{Z}_+$), or relative to a linear order $\preceq$ on $\mathbb{Z}^d$, i.e.\ $H = \{j\in\mathbb{Z}^d : 0\preceq j\}$. We consider the problem of interpolation at the points of $H$ from the space of series expansions in terms of the $H$-shifts of a decaying kernel $ϕ$. Using the Wiener-Hopf factorization of the symbol for cardinal interpolation with $ϕ$ on $\mathbb{Z}^d$, we derive some essential properties of semi-cardinal interpolation on $H$, such as exi…

将径向基函数插值的误差界扩展到高阶Sobolev范数下的误差度量

Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms

本文将径向基函数插值的加倍技巧扩展至高阶Sobolev范数误差度量，适用于满足易验证假设的核族，并获得了新收敛率与Bernstein估计。

超越原生空间的核逼近——在流形逼近中的应用

Kernel approximation beyond the native space -- with applications to approximation on manifolds

针对嵌入流形上的正定和条件正定核，提出一种推广的逼近方案，通过积分算子范围刻画目标函数，并建立新的Bernstein不等式以实现原生空间紧嵌入Sobolev空间中的插值误差估计。