Thomas Hangelbroek

Localized Bases for Kernel Spaces on the Unit Sphere

Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data and is central to many meshless methods. For a set of

Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant

N

Kernel based quadrature on spheres and other homogeneous spaces

scattered sites, the standard basis for such a space utilizes

Kernel Approximation on Manifolds II: The

N

L_{\infty}

globally supported kernels; computing with it is prohibitively expensive for large

Norm of the

N

L_2

. Easily computable, well-localized bases with “small-footprint” basis elements---i.e., elements using only a small number of kernels---have been unavailable. Working on

Projector

\mathbb{S}^2

Nonlinear approximation using Gaussian kernels

, with focus on the restricted surface spline kernels (e.g., the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only

Cardinal Interpolation with Gaussian Kernels

\mathcal{O}((\log N)^2)