Related papers: $n$-widths and Approximation theory on Compact Rie…
We establish simultaneous approximation properties of weighted first-order Sobolev orthogonal projectors onto spaces of polynomials of bounded total degree in the Euclidean unit ball. The simultaneity is in the sense that we provide bounds…
We study certain weighted Bergman and weighted Besov spaces of holomorphic functions in the polydisk and in the unit ball. We seek Mergelyan-type conditions on the non-radial weight function to guarantee that the dilations of a given…
We say a map f:X \to Y is an \epsilon-embedding if it is continuous and the diameter of the fibres is less than \epsilon. This type of maps is used in the notion of Urysohn width (sometimes referred to as Alexandrov width), a_n(X). It is…
In this paper we establish commmutator estimates for the Dirichlet-to-Neumann Map associated to a divergence form elliptic operator in the upper half-space $\mathbb{R}^{n+1}_+:=\{(x,t)\in \mathbb{R}^n \times (0,\infty)\}$, with uniformly…
The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$,…
We obtain estimates of commutators of singular integral operators in Lipschitz spaces and apply the results to boundary regularity of elliptic equations in the plane. We obtain an explicit asymptotic formula for the Bergman projection.
Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an…
Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of…
We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal.,…
We study uniform $\epsilon-$BPB approximations of bounded linear operators between Banach spaces from a geometric perspective. We show that for sufficiently small positive values of $\epsilon,$ many geometric properties like smoothness,…
We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame coefficients, using the frames we have constructed in [8].
We study the asymptotics of almost holomorphic sections $s \in H^0_J(M, \omega)$ of an ample line bundle $L \to M$ over an almost complex symplectic manifold in the sense of Boutet de Monvel-Guillemin. Such sections are defined as the…
We study nonlinear n-term approximation of harmonic functions on the unit ball in $R^d$ from linear combinations of shifts of the Newtonian kernel (fundamental solution of the Laplace equation) in BMO. A sharp Jackson estimate is…
Non-asymptotic bounds for Gaussian and bootstrap approximation have recently attracted significant interest in high-dimensional statistics. This paper studies Berry-Esseen bounds for such approximations with respect to the multivariate…
For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em metric…
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…
We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based on the L\"owner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a…
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there…
In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the…
A rather complete investigation of anisotropic Bessel potential, Besov, and H\"older spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends'…