Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of M, ∆Mf (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations ∆hn,nf(p)− 1 |μ|∆Mf(p) for smooth functions f bo...

Combining information both within and across trajectories, we propose a simple estimator for the local regularity of trajectories stochastic process. Independent are measured with errors at randomly sampled time points. The proposed approach is model-free applies to large class processes. Non-asymptotic bounds concentration derived. Given estimate regularity, build nearly optimal polynomial smo...

We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator (L,D(L)) in the space of all uniformly continuous and bounded functions f : H → R, where H is a separable Hilbert space. We consider a perturbation of the form N0φ = Lφ + 〈Dφ,F 〉 where F : H → H is bounded and Fréchet differentiable with uniformly continuous and bounded differential. Hence, we prove that N...

We prove that in Orlicz spaces endowed with Orlicz norm the uniformly normal structure is equivalent to the reflexivity.

In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values Banach spaces.

Recently, in [1], it has been proved that if B is an open unit ball in a Cartesian product l2 × l2 furnished with the lp-norm ‖ · ‖ and kB is the Kobayashi distance on B, then the metric space (B,kB) is locally uniformly convex in linear sense. Our construction of domains, which are locally uniformly convex in their Kobayashi distances, is based on the ideas from [1]. Such domains play an impor...

Let K ⊂ Rmn be a compact and convex set of m×n matrices and let {uj} be a sequence in W 1,1 loc (Rn;Rm) that converges to K in the mean, i.e. ∫ Rn dist(Duj , K) → 0. I show that there exists a sequence vj of Lipschitz functions such that ‖dist(Dvj , K)‖∞ → 0 and L({uj 6= vj}) → 0. This refines a result of Kewei Zhang (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 313-326), who showed that...