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Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is…
We consider nonparametric regression with functional covariates, that is, they are elements of an infinite-dimensional Hilbert space. A locally polynomial estimator is constructed, where an orthonormal basis and various tuning parameters…
We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with…
Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on…
Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…
In this paper, we consider approximations of principal component projection (PCP) without explicitly computing principal components. This problem has been studied in several recent works. The main feature of existing approaches is viewing…
The article is devoted to comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations in the framework of the method of…
This note mainly concerns the binomial power function, defined as $(1+x^q)^{r}$. We construct systems of polynomials related to non-local approximation, which allows us to establish the density results on $C[a,b]$, where $a,b\in\mathbb{R}$.…
The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there…
Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…
Derivative-matching approximations are constructed as power series built from functions. The method assumes the knowledge of special values of the Bell polynomials of the second kind, for which we refer to the literature. The presented…
The vector space of all polynomial functions of degree $k$ on a box of dimension $n$ is of dimension ${n \choose k}$. A consequence of this fact is that a function can be approximated on vertices of the box using other vertices to higher…
This paper proposes a novel localized Fourier extension method for approximating non-periodic functions via domain segmentation. By partitioning the computational domain into subregions with uniform discretization scales, the method…
We make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on \cite{bigbook}. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network…
We present a dimension-incremental algorithm for the nonlinear approximation of high-dimensional functions in an arbitrary bounded orthonormal product basis. Our goal is to detect a suitable truncation of the basis expansion of the…
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…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
A landmark result from rational approximation theory states that $x^{1/p}$ on $[0,1]$ can be approximated by a type-$(n,n)$ rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev…
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…
A mathematical proposition with a trainable pair, operator and quantum circuit, are introduced to approximate functions expressed as cubic Taylor polynomials, numerical simulations illustrate three cases.