Related papers: Fully discrete needlet approximation on the sphere

This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify…

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost…

We propose an extrapolation technique that allows accuracy improvement of the discrete dipole approximation computations. The performance of this technique was studied empirically based on extensive simulations for 5 test cases using many…

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold $M$ when given a sample on a finite point set. We prove that the quality of the sample is given by the $L_\gamma(M)$-average of the geodesic…

A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev measure in the $d$-cube with $\approx n^d/2^{d-1}$ nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree $n$ in three…

Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for $L_2$ -approximation of Sobolev functions into estimates for $L_\infty$-approximation, with precise control of all…

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator…

We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written…

This paper considers the following question: how well can depth-two ReLU networks with randomly initialized bottom-level weights represent smooth functions? We give near-matching upper- and lower-bounds for $L_2$-approximation in terms of…

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of…

This paper proves that given a doubling weight $w$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$, there exists a positive constant $K_w$ such that for each positive integer $n$ and each integer $N\geq \max_{x\in \mathbb{S}^{d-1}}…

The paper establishes the strong convergence rates of a spatio-temporal full discretization of the stochastic wave equation with nonlinear damping in dimension one and two. We discretize the SPDE by applying a spectral Galerkin method in…

The Dirichlet prior is widely used in estimating discrete distributions and functionals of discrete distributions. In terms of Shannon entropy estimation, one approach is to plug-in the Dirichlet prior smoothed distribution into the entropy…

We consider error estimates for the fully discretized instationary Navier-Stokes problem. For the spatial approximation we use conforming inf-sup stable finite element methods in conjunction with grad-div and local projection stabilization…

We consider an initial- and Dirichlet boundary- value problem for a fourth-order linear stochastic parabolic equation, in two or three space dimensions, forced by an additive space-time white noise. Discretizing the space-time white noise a…

In this paper, we introduce a new semi-discrete modulus of smoothness, which generalizes the definition given by Kolomoitsev and Lomako (KL) in 2023 (in the paper published in the J. Approx. Theory), and we establish very general one- and…

The approximation of integral type functionals is studied for discrete observations of a continuous It\^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions…

The matrix spectral and nuclear norms appear in enormous applications. The generalizations of these norms to higher-order tensors is becoming increasingly important but unfortunately they are NP-hard to compute or even approximate. Although…

In this paper, we develop a numerical multiscale method to solve elliptic boundary value problems with heterogeneous diffusion coefficients and with singular source terms. When the diffusion coefficient is heterogeneous, this adds to the…

We analyze the accuracy of the discrete least-squares approximation of a function $u$ in multivariate polynomial spaces $\mathbb{P}_\Lambda:={\rm span} \{y\mapsto y^\nu \,: \, \nu\in \Lambda\}$ with $\Lambda\subset \mathbb{N}_0^d$ over the…