Related papers: The Error in Multivariate Linear Extrapolation wit…

The problem of extrapolation and interpolation of asymptotic series is considered. Several new variants of improving the accuracy of the self-similar approximants are suggested. The methods are illustrated by examples typical of chemical…

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known…

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…

Single hidden layer feedforward neural networks can represent multivariate functions that are sums of ridge functions. These ridge functions are defined via an activation function and customizable weights. The paper deals with best…

In this paper, we propose a novel extrapolation coefficient scheme within a new extrapolation term and develop an accelerated proximal gradient algorithm. We establish that the algorithm achieves a sublinear convergence rate. The proposed…

We examine the necessity of interpolation in overparameterized models, that is, when achieving optimal predictive risk in machine learning problems requires (nearly) interpolating the training data. In particular, we consider simple…

We study the bias-variance tradeoff within a multiscale approximation framework. Our approach uses a given quasi-interpolation operator, which is repeatedly applied within an error-correction scheme over a hierarchical data structure. We…

This paper describes applications of extrapolation for the computation of coefficients in an expansion of infrared divergent integrals. An extrapolation procedure is performed with respect to a parameter introduced by dimensional…

One classical measure of the quality of an interpolating function is its Lipschitz constant. In this paper we consider interpolants with additional smoothness requirements, in particular that their derivatives be Lipschitz. We show that…

Unitary best approximation to the exponential function on an interval on the imaginary axis has been introduced recently. In the present work two algorithms are considered to compute this best approximant: an algorithm based on rational…

Explicit pointwise error bounds for the interpolation of a smooth function by piecewise exponential splines of order four are given. Estimates known for cubic splines are extended to a natural class of piecewise exponential splines which…

Interpolation by various types of splines is the standard procedure in many applications. In this paper we shall discuss harmonic spline "interpolation" (on the lines of a grid) as an alternative to polynomial spline interpolation (at…

The main contribution of this paper is twofold: On the one hand, a general framework for performing Hermite interpolation on Riemannian manifolds is presented. The method is applicable, if algorithms for the associated Riemannian…

This paper addresses the problem of approximating a function of bounded variation from its scattered data. Radial basis function(RBF) interpolation methods are known to approximate only functions in their native spaces, and to date, there…

The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing…

The Performance Estimation Problem methodology makes it possible to determine the exact worst-case performance of an optimization method. In this work, we generalize this framework to first-order methods involving linear operators. This…

Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…

We present the error analysis of Lagrange interpolation on triangles. A new \textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on…

We introduce remarkable upper bounds for the interpolation error constants on triangles, which are sharp and given by simple formulas. These constants are crucial in analyzing interpolation errors, particularly those associated with the…

Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the…