64 citations · 228 across the 21 of their papers we have counts for
22 papers
Smoothing by, and eccentric smoothing of, compactly supported RBFs
Thomas Hangelbroek, Christian Rieger
We consider compactly supported RBFs having algebraically decaying Fourier transforms. Here we focus especially on generalized Wendland RBFs and their modification by making them s…
Kernel approximation beyond the native space -- with applications to approximation on manifolds
Thomas Hangelbroek, Christian Rieger, Grady B. Wright
This article treats kernel approximation and interpolation on embedded manifolds of using restrictions of positive and conditionally positive definite kernels. The ma…
A Semi-Lagrangian scheme on embedded manifolds using generalized local polynomial reproductions
Thomas Hangelbroek, Christian Rieger, Grady B. Wright
We analyze rates of uniform convergence for a class of high-order semi-Lagrangian schemes for first-order, time-dependent partial differential equations on embedded submanifolds of…
Generalized local polynomial reproductions
Thomas Hangelbroek, Christian Rieger, Grady B. Wright
We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polyno…
Spectral stability and perturbation results for kernel differentiation matrices on the sphere
Thomas Hangelbroek, Christian Rieger, Grady Wright
We investigate the spectrum of differentiation matrices for certain operators on the sphere that are generated from collocation at a set of scattered points with positive defin…
Kernel Multi-Grid on Manifolds
Thomas Hangelbroek, Christian Rieger
Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if…
Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms
Thomas Hangelbroek, Christian Rieger
Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based…
Anisotropic Gaussian approximation in
Wolfgang Erb, Thomas Hangelbroek, Amos Ron
Let be the dictionary of Gaussian mixtures: the functions created by affine change of variables of a single Gaussian in dimensions. is used pervasiv…
On a polyharmonic Dirichlet problem and boundary effects in surface spline approximation
Thomas C. Hangelbroek
For compact domains with smooth boundaries, we present an approximation scheme for surface spline approximation that delivers precise approximation orders on well known smoot…
An inverse theorem for compact Lipschitz regions in using localized kernel bases
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger +1
While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important…
Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger +1
This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approxi…
The Polyharmonic Dirichlet Problem and Path Counting
Thomas Hangelbroek, Aaron Lauve
The purpose of this article is to provide a solution to the -fold Laplace equation in the half space under certain Dirichlet conditions. The solutions we present are a s…
Kernel Based Quadrature on Spheres and Other Homogeneous Spaces
E. Fuselier, T. Hangelbroek, F. J. Narcowich +2
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research…
Localized bases for kernel spaces on the unit sphere
E. Fuselier, T. Hangelbroek, F. J. Narcowich +2
Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered…
Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation
T. Hangelbroek, F. J. Narcowich, J. D. Ward
This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected Riemanni…
Cardinal Interpolation with Gaussian Kernels
Thomas Hangelbroek, Wolodymyr Madych, F. J. Narcowich +1
In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes Sobolev error estimates and shows that the error i…
Nonlinear Approximation Using Gaussian Kernels
Thomas Hangelbroek, Amos Ron
It is well-known that non-linear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but…
Surface Spline Approximation on SO(3)
Thomas Hangelbroek, Dominik Schmid
The purpose of this article is to introduce a new class of kernels on SO(3) for approximation and interpolation, and to estimate the approximation power of the associated spaces. T…
The Penalized Lebesgue Constant for Surface Spline Interpolation
Thomas Hangelbroek
Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying densi…
On Local RBF Approximation
Thomas Hangelbroek
The purpose of this paper is to investigate RBF approximation with highly nonuniform centers. Recently, DeVore and Ron have developed a notion of the local density of a set of cent…
Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
Thomas Hangelbroek, Fran J. Narcowich, Joe D. Ward
The purpose of this paper is to establish that for any compact, connected C^{\infty} Riemannian manifold there exists a robust family of kernels of increasing smoothness that are w…
Polyharmonic approximation on the sphere
Thomas Hangelbroek
The purpose of this article is to provide new error estimates for a popular type of SBF approximation on the sphere: approximating by linear combinations of Green's functions of po…