Localized linear polynomial operators and quadrature formulas on the sphere

The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere \SS^q ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.Comment: 24 pages 2 figures, accepted for publication in SIAM J. Numer. Ana

Marcinkiewicz--Zygmund measures on manifolds

Let ${\mathbb X}$ be a compact, connected, Riemannian manifold (without boundary), $\rho$ be the geodesic distance on ${\mathbb X}$, $\mu$ be a probability measure on ${\mathbb X}$, and $\{\phi_k\}$ be an orthonormal system of continuous functions, $\phi_0(x)=1$ for all $x\in{\mathbb X}$, $\{\ell_k\}_{k=0}^\infty$ be an nondecreasing sequence of real numbers with $\ell_0=1$, $\ell_k\uparrow\infty$ as $k\to\infty$, $\Pi_L:={\mathsf {span}}\{\phi_j : \ell_j\le L\}$, $L\ge 0$. We describe conditions to ensure an equivalence between the $L^p$ norms of elements of $\Pi_L$ with their suitably discretized versions. We also give intrinsic criteria to determine if any system of weights and nodes allows such inequalities. The results are stated in a very general form, applicable for example, when the discretization of the integrals is based on weighted averages of the elements of $\Pi_L$ on geodesic balls rather than point evaluations.Comment: 28 pages, submitted for publicatio