相关论文: An essay on some problems of approximation theory

The Universal Approximation Theorem posits that neural networks can theoretically possess unlimited approximation capacity with a suitable activation function and a freely chosen or trained set of parameters. However, a more practical…

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…

We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an…

In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Gr\"unwald derivatives in $L_p$, $0<p<1$. We extend known inequalities to the whole range of…

This review paper highlights the main aspects of the development of research related to the solution of extreme problems in the theory of approximation in the spaces ${\mathcal S}^p$ and $B{\mathcal S}^p$ of periodic and almost periodic…

We show optimal FPT-approximability results for solving almost satisfiable systems of modular linear equations, completing the picture of the parameterized complexity and FPT-approximability landscape for the Min-$r$-Lin$(\mathbb{Z}_m)$…

Let $ f(z)=\int(z-x)^{-1}d\mu(x) $, where $ \mu $ is a Borel measure supported on several subintervals of $ (-1,1) $ with smooth Radon-Nikodym derivative. We study strong asymptotic behavior of the error of approximation $ (f-r_n)(z) $,…

The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our…

In this paper, we examine the problem of approximating a general linear dimensionality reduction (LDR) operator, represented as a matrix $A \in \mathbb{R}^{m \times n}$ with $m < n$, by a partial circulant matrix with rows related by…

We prove two results concerning an Ulam-type stability problem for homomorphisms between lattices. One of them involves estimates by quite general error functions; the other deals with approximate (join) homomorphisms in terms of certain…

Given an arbitrary finite set of data F= {f_1,..., f_m} in L2(Rd) we prove the existence and show how to construct a "small shift invariant space" that is "closest" to the data F over certain class of closed subspaces of L2(Rd). The…

This paper discusses a general and useful stability principle which, roughly speaking, says that given a uniformly continuous function defined on an arbitrary metric space, if the function is bounded on the constraint set and we slightly…

Estimating the density of a distribution from its samples is a fundamental problem in statistics. Hypothesis selection addresses the setting where, in addition to a sample set, we are given $n$ candidate distributions -- referred to as…

The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to [2,\infty)$, we determine…

For given $p\in\lbrack1,\infty]$ and $g\in L^{p}\mathbb{(R)}$, we establish the existence and uniqueness of solutions $f\in L^{p}(\mathbb{R)}$, to the equation \[ f(x)-af(bx)=g(x), \] where $a\in\mathbb{R}$, $b\in\mathbb{R} \setminus…

In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements,…

By application of the theory for second-order linear differential equations with two turning points developed in \cite{Olver1975}, uniform asymptotic approximations are obtained for the Lam\'{e} and Mathieu functions with a large real…

This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including {optimal feedback…

The celebrated universal approximation theorems for neural networks roughly state that any reasonable function can be arbitrarily well-approximated by a network whose parameters are appropriately chosen real numbers. This paper examines the…

We consider a Fermi-Pasta-Ulam-Tsingou lattice with randomly varying coefficients. We discover a relatively simple condition which when placed on the nature of the randomness allows us to prove that small amplitude/long wavelength solutions…