Related papers: An essay on some problems of approximation theory

Universal Approximation Theorems establish the density of various classes of neural network function approximators in $C(K, \mathbb{R}^m)$, where $K \subset \mathbb{R}^n$ is compact. In this paper, we aim to extend these guarantees by…

We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden…

In arXiv:1306.2914 a method for approximate solution of Sturm-Liouville equations and related spectral problems was presented based on the construction of the Delsarte transmutation operators. The problem of numerical approximation of…

In applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements. Series method offers that by approximating the unknown function based on $k$ basis…

Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…

Many practical problems need the output of a machine learning model to satisfy a set of constraints, $K$. Nevertheless, there is no known guarantee that classical neural network architectures can exactly encode constraints while…

We study the polynomial approximation problem in $L^2(\mu_1)$ where $\mu_1(dx) = e^{-|x|}/2 dx$. We show that for any absolutely continuous function $f$, $$ \sum_{k=1}^{\infty} \log^2(e+k) \langle f, P_k \rangle^2 \ \leq C \left(…

The approximation of a general $d$-variate function $f$ by the shifts $\phi(\cdot-\xi)$, $\xi\in\Xi\subset \Rd$, of a fixed function $\phi$ occurs in many applications such as data fitting, neural networks, and learning theory. When…

Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first…

Given a function dictionary $\cal D$ and an approximation budget $N\in\mathbb{N}^+$, nonlinear approximation seeks the linear combination of the best $N$ terms $\{T_n\}_{1\le n\le N}\subseteq{\cal D}$ to approximate a given function $f$…

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show…

It was pointed out to us that the proof of a crucial lemma (Lemma 5.3) in the paper is incorrect. Thus the approximation theorem (Theorem 0.1) for L^2 torsion of an amenable covering of a finite simplicial complex remains unproved. However,…

A method is suggested for treating those complicated physical problems for which exact solutions are not known but a few approximation terms of a calculational algorithm can be derived. The method permits one to answer the following rather…

When an Approximation Theorist looks at well-posed PDE problems or operator equations, and standard solution algorithms like Finite Elements, Rayleigh-Ritz or Trefftz techniques, methods of fundamental or particular solutions and their…

Finite (word) state transducers extend finite state automata by defining a binary relation over finite words, called rational relation. If the rational relation is the graph of a function, this function is said to be rational. The class of…

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire…

Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from…

We study integration and $L^2$-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure…

Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large…