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 well suited for interpolation. They generate Lagrange functions that are uniformly bounded and decay away from their center at an exponential rate. An immediate corollary is that the corresponding Lebesgue constant will be uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The analysis needed for these results was inspired by some fundamental work of Matveev where the Sobolev decay of Lagrange functions associated with certain kernels on \Omega \subset R^d was obtained. With a bit more work, one establishes the following: Lebesgue constants associated with surface splines and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi are quasi-uniformly distributed. The non-Euclidean case is more involved as the geometry of the underlying surface comes into play. In addition to establishing bounded Lebesgue constants in this setting, a "zeros lemma" for compact Riemannian manifolds is established.Comment: 33 pages, 2 figures, new title, accepted for publication in SIAM J. on Math. Ana

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 N scattered sites, the standard basis for such a space utilizes N globally supported kernels; computing with it is prohibitively expensive for large N . 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 S 2 , 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 O((log N ) 2 ) kernels, which makes the construction computationally cheap. We prove that the new basis is Lp stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal Lp approximation orders. Although our focus is on S 2 , much of the theory applies to other manifolds-S d , the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on S 2 involving over 150,000 data sites.

It is well known that nonlinear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron (in press) [2] for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore-Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.

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. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate -optimally so in many cases -, and stable under an increasing number of nodes and in the presence of noise, provided the set X of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to S n , oblate spheroids for instance. The weights are obtained by solving a single linear system. For S 2 , and the restricted thin plate spline kernel r 2 log r, these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.

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 C ∞ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for L 2 minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.

This article addresses two topics of significant mathematical and practical interest in the theory of kernel approximation: the existence of local and stable bases and the L p -boundedness of the least squares operator. The latter is an analogue of the classical problem in univariate spline theory, known there as the "de Boor conjecture". A corollary of this work is that for appropriate kernels the least squares projector provides universal near-best approximations for functions f ∈ L p , 1 ≤ p ≤ ∞.

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 and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the Lp norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Matérn and surface spline RBFs). In this way it extends the boundary-free construction recently presented in [8].where in defining ρ(X, D) we assume that q(X) > 0.When there is no chance of confusion, we drop dependence in these parameters on X and D (referring simply to h, q and ρ).Remark 2.1. A finite fill distance h guarantees that the set D is covered by the family of balls B(ξ, h) := {x ∈ D | dist(x, ξ) < h}, ξ ∈ X. A positive separation radius q guarantees that B(ξ, q) ∩ B(ζ, q) = {} for distinct ζ, ξ ∈ Ξ. The mesh ratio, which automatically satisfies ρ ≥ 1, measures the uniformity of the distribution of X in D. The larger ρ(X, D) is, the less uniform the distribution is. If ρ is finite, then we say that the point set X is quasi-uniformly distributed (in D), or simply that X is quasi-uniform.Note that, for a compact subset D and a nonempty, finite subset X ⊂ D, the fill distance and separation radius are both positive and finite 0 < q < h < ∞. Consequently, ρ is finite, too.Many of the results in this article depend in some way on the geometry of the point set X -often this emerges in an estimate, where a constant depends on ρ. In most cases, (as one may expect) the strength of the estimate degrades as ρ increases. Throughout the paper, we have attempted make this control explicit, by factoring, whenever possible, the constant into a part which is totally independent of the point set, and another, which is a function of ρ.

In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes L p Sobolev error estimates and shows that the error is controlled by the L p multiplier norm of a Fourier multiplier closely related to the cardinal interpolant, and comparable to the Hilbert transform. Consequently, its multiplier norm is bounded independent of the grid spacing when 1 < p < ∞, and involves a logarithmic term when p = 1 or ∞.