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

A unified framework for harmonic analysis of functions on directed graphs and changing data

We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner

Eignets for function approximation on manifolds

Let \XX be a compact, smooth, connected, Riemannian manifold without boundary, G:\XX\times\XX\to \RR be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form $\sum_{j=1}^M a_jG(\circ,y_j)$, where a_j\in\RR, y_j\in\XX, $1\le j\le M$. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in L^p(\mu;\XX) for a general class of measures $\mu$ and kernels $G$. Our algorithm yields linear operators. Using the minimal separation amongst the centers $y_j$ as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every \emph{individual function}. We also give estimates on the coefficients $a_j$ in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.Comment: 28 pages. Articles in press; Applied and Computational Harmonic Analysis, 200