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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 $\mathbb{S}^n$, oblate spheroids for instance. The weights are obtained by solving a single linear system. For $\mathbb{S}^2$, 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., Comment: 31 pages, 8 figures

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 $N$ \emph{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 $\sphere$, 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 $\mathcal{O}((\log N)^2)$ kernels, which makes the construction computationally cheap. We prove that the new basis is $L_p$ 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 $L_p$ approximation orders. Although our focus is on $\mathbb{S}^2$, much of the theory applies to other manifolds - $\mathbb{S}^d$, 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 $\mathbb{S}^2$ involving over one hundred fifty thousand data sites., Comment: This article supersedes arXiv:1111.1013 "Better bases for kernel spaces," which proved existence of better bases for various kernel spaces. This article treats a smaller class of kernels, but presents an algorithm for constructing better bases and demonstrates its effectiveness with more elaborate examples. A quasi-interpolation scheme is introduced that provides optimal linear convergence rates

In this article we investigate the feasibility of constructing stable, local bases for computing with kernels. In particular, we are interested in constructing families $(b_{\xi})_{\xi\in\Xi}$ that function as bases for kernel spaces $S(k,\Xi)$ so that each basis function is constructed using very few kernels. In other words, each function $b_{\zeta}(x) = \sum_{\xi\in\Xi} A_{\zeta,\xi} k(x,\xi)$ is a linear combination of samples of the kernel with few nonzero coefficients $A_{\zeta,\xi}$. This is reminiscent of the construction of the B-spline basis from the family of truncated power functions. We demonstrate that for a large class of kernels (the Sobolev kernels as well as many kernels of polyharmonic and related type) such bases exist. In fact, the basis elements can be constructed using a combination of roughly $O(\log N)^d$ kernels, where $d$ is the local dimension of the manifold and $N$ is the dimension of the kernel space (i.e. $N=#\Xi$). Viewing this as a preprocessing step -- the construction of the basis has computational cost $O(N(\log N)^d)$. Furthermore, we prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of $\Xi$., Comment: 26 pages, 5 figures, 3 tables

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 $C^\infty$ 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., Comment: Final manuscript. Contains a new zeros lemma for regions on compact Riemannian manifolds with Lipschitz boundaries -- constants are in terms of parameters from a cone condition. 2 figures. 41 pages