Marcinkiewicz--Zygmund measures on manifolds

Abstract

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 $\{{\varphi }_k\}$ be an orthonormal system of continuous functions, ${\varphi }_0(x)=1$ 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 $k\to \infty$, ${\Pi }_L:=\mathsf{span}\{{\varphi }_j:{\ell }_j\le L\}$, $L\ge 0$. 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.

28 pages, submitted for publication

References in corpus (2)

• Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian
• Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds