Related papers: Hyperinterpolation beyond exact cubature: a spectr…
Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproducing kernel Hilbert space. Beyond this classical setting,…
We develop and analyze a class of matrix-valued spherical-convolution kernels stemming from scaled zonal functions on $\mathbb{S}^2,$ the unit sphere embedded in $\mathbb{R}^3$. The construct of these kernels utilizes the Legendre…
We propose, analyze, and implement interpolatory approximations and Filon-type cubature for efficient and accurate evaluation of a class of wideband generalized Fourier integrals on the sphere. The analysis includes derivation of (i)…
This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree $n$ via hyperinterpolation. Hyperinterpolation of degree $n$ is a discrete approximation of the $L^2$-orthogonal…
This paper studies a generalization of hyperinterpolation over the high-dimensional unit cube. Hyperinterpolation of degree \( m \) serves as a discrete approximation of the \( L_2 \)-orthogonal projection of the same degree, using Fourier…
Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that…
We design a quasi-interpolation operator from the Sobolev space $H^1_0(\Omega)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1)…
We establish a deterministic and stochastic spherical quasi-interpolation framework featuring scaled zonal kernels derived from radial basis functions on the ambient Euclidean space. The method incorporates both quasi-Monte Carlo and Monte…
We derive new estimates for distances between optimal matchings of eigenvalues of non-normal matrices in terms of the norm of their difference. We introduce and estimate a hyperbolic metric analogue of the classical spectral-variation…
We consider quasi-interpolation with a main application in radial basis function approximations and compression in this article. Constructing and using these quasi-interpolants, we consider wavelet and compression-type approximations from…
This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces.This operator gives optimal estimates of the…
In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a…
We introduce an interpolation--regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed…
Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic…
The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons. In the numerical section we…
A full interpolation theory for Sobolev functions with smoothness between 0 and 1 and vanishing trace on a part of the boundary of an open set is established. Geometric assumptions are of mostly measure theoretic nature and reach beyond…
We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to…
Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a…
High dimensional integrals can be approximated well by quasi-Monte Carlo methods. However, determining the number of function values needed to obtain the desired accuracy is difficult without some upper bound on an appropriate semi-norm of…
In computational practice, we often encounter situations where only measurements at equally spaced points are available. Using standard polynomial interpolation in such cases can lead to highly inaccurate results due to numerical…