Uniform Approximation of Continuous Couplings

The quantity of the precise upper bound of the deviations of the linear methods of summation, which are determined by rectangular number matrix Λ = IIλn,kII on the classes of continuous periodic functions in the uniform metric is given. As possible application of the obtained results, we study the asymptotic behavior of threeharmonic Poisson integrals in the case when the classes W r∞, r ∈ N , are an object of approximation. The asymptotic equalities reveal the theoretical foundations and mathematical features of one of the main problems of approximation theory − the Kolmogorov-Nikol'sky problem. In particular, the problem is solved for the threeharmonic Poisson integrals on the Sobolev classes in the uniform metric. It is found that the threeharmonic Poisson integrals possess approximation properties that are different from the properties of the harmonic and biharmonic Poisson integrals, which were studied previously, and some concepts and techniques of approximation theory can also be useful in studying the spaces of functions with generalized derivatives. An important moment in the solution of this problem is the fact that with the help of the asymptotic equalities, which are studied, a wide range of economic problems can be solved, the solution of which by methods of classical linear algebra and mathematical analysis is a complicated process. Economic modeling and forecasting on the basis of the constructed mathematical model can be used in the analysis of processes of economic dynamics, considering polyharmonic regimes. The purpose of the work is to develop a mathematical apparatus that allows to build mathematical models of periodic economic processes. Modeling serves as a means of analyzing the economy and the phenomena occurring in it, as well as justifying the decisions made, forecasting and managing economic processes and objects. We also analyze some fundamental problem of the modern economy, solved by methods of the approximation theory

Одним з напрямкiв наукових дослiджень Ю. В. Козаченка є рiвняння математичної фiзики з випадковими факторами. Цi фактори можуть мати рiзну природу: випадковi початковi умови, випадковi крайовi умови, випадкова права частина, випадковi коефiцiєнти i т. д. Умови та оцiнки збiжностi за ймовiрнiстю випадкових рядiв знаходять широке застосування при розв’язаннi задач математичної фiзики з випадковими умовами. Фiзичнi постановки таких задач розглядав Кампе де Фер’є . Вiн розглядав крайову задачу для рiвняння коливання струни з випадковими початковими умовами. У роботах В. В. Булдигiна показано, що вимога, щоб майже всi реалiзацiї випадкової початкової функцiї задовольняли умови, при яких є розв’язуваною детермiнована задача, значно звужує клас випадкових умов, за яких розв’язок iснує в класичному розумiннi. Є багато робiт, в яких вивчались задачi математичної фiзики з випадковими умовами, якi базуються на дослiдженнi збiжностi за ймовiрностю в функцiональних просторах послiдовностi випадкових функцiй, що апроксимують розв’язки крайових задач. Зауважимо, що у бiльшостi з цих робiт, для знаходження умов рiвномiрної збiжностi випадкових рядiв застосовується метод, що ґрунтується на iдеї Ж.Канаха. Булдигiним В.В. та Козаченком Ю.В. був запропонований метод, який дозволяє обґрунтовувати застосування методу Фур’є до задач математичної фiзики у багатовимiрному випадку. Метод, що ґрунтується на iдеї Кахана для цього випадку не пiдходить. У роботах Козаченка Ю.В. та його учнiв дослiджувалися рiвняння гiперпболiчного та параболiчного типiв математичної фiзики з випадковими факторами. Зокрема, вивчалися властивостi класичних та узагальнених розв’язкiв таких задач, було обґрунтувано застосування методу Фур’є, знайдено оцiнок для розподiлу супремуму розв’язкiв, та побудовано моделi розв’язкiв деяких задач, що наближають розв’язок iз заданою надiйнiстю та точнiстю в рiвномiрнiй метрицi. Всi цi результати мають не лише теоретичне, але й практичне застосування для подальшого вивчення та розвинення теорiї гiперболiчних i параболiчних рiвнянь математичної фiзики з випадковими факторами. Крiм того, цi результати дозволяють моделювати розв’язки крайових задач математичної фiзики iз заданою надiйнiстю та точнiстю в рiвномiрнiй метрицi, що може застосовуватися в наукових дослiдженнях в галузi радiотехнiки, фiзики, геофiзики, фiнансової математики, математичної економiки, в технiчних науках та в механiцi, зокрема, де використовуються методи комп’ютерного моделювання випадкових процесiв.

Uniform convergence of regularization methods for linear ill-posed problems

Wiener's tauberian theorems for the Fourier–Bessel transformation and uniform approximation by RBF networks of Delsarte translates

Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems

The approximation characteristics of trigonometric sums Un,pψ of special type on the class Cβ,∞ψ of (ψ,β)-differentiable (in the sense of A. I. Stepanets) periodical functions are studied. Because of agreement between parameters of approximative sums and approximated classes, the solution of Kolmogorov-Nikol’skii problem is obtained in a sufficiently general case. It is shown that in a number of important cases these sums provide higher order of approximation in comparison with Fourier sums, de la Vallée Poussin sums, and others on the class Cβ,∞ψ in the uniform metric. The range of parameters in which the sums Un,pψ give the order of the best uniform approximation on the classes Cβ,∞ψ is indicated.

A semi-analytical approximation of the normal derivative of the simple layer heat potential near the boundary of a two-dimensional domain with 𝐶 5 smoothness is proposed. The calculation of the integrals that arise after piecewise quadratic interpolation of the density function with respect to the variable of arc length 𝑠, is carried out using analytical integration over the variable 𝜌 = √ 𝑟 2 − 𝑑 2 , where 𝑟 and 𝑑 are the distances from the observation point to the integration point and to the boundary of the domain, respectively. To do this, the integrand is represented as the sum of two products, each of which consists of two factors, namely: a function smooth in а near-boundary domain containing the Jacobian of the transition from the integration variable 𝑠 to the variable 𝜌, and a weight function containing a singularity at 𝑟 = 0 and uniformly absolutely integrable in the near-boundary region. Smooth functions are approximated with the help of the piecewise quadratic interpolation over the variable 𝜌, and then analytical integration becomes possible. Analytical integration over 𝜌 is carried out on a section of the boundary fixed in width, containing the projection of the observation point, and on the rest of the boundary, the integrals over 𝑠 are calculated using the Gauss formulas. Integration over the parameter of 𝐶 0 -semigroup formed by time shift operators is also carried out analytically. To do this, the 𝐶 0 -semigroup is approximated using the piecewise quadratic interpolation over its parameter. It is proved that the proposed approximations have stable cubic convergence in the Banach space of continuous functions with the uniform norm, and such convergence is uniform in the closed near-boundary region. The results of computational experiments on finding of the normal derivative of solutions of the second initial-boundary problem of heat conduction in a unit circle with a zero initial condition are presented, confirming the uniform cubic convergence of the proposed approximations of the normal derivative of the simple layer heat potential.

Paralleling the classical strong uniqueness, in this paper we consider the modified strong uniqueness which measures the distance between the best approximation and the achieved approximation in the parameter norm instead of uniform (function) norm in which the classical strong uniqueness measures the distance. We introduce a quantity called modified strong uniqueness constant which can be used to bound the distance mentioned above (if we can bound the difference between the minimal approximation error norm and the achieved approximation error norm), and deduce a computation formula for it for both linear and nonlinear uniform approximations. Cline's arguments for the classical strong uniqueness constant are used, but we make some modifications due to the turning of our attention from uniform (function) norm to parameter norm. Implementation of the computation, and examples for computing both classical and modified strong uniqueness constants are given. We also introduce a quantity which is analogous to Lipschitz constant.

We study approximations of functions byn-analytic polynomials in the uniform norm on closed rectifiable Jordan curves in the complex plane. It is shown that, in contrast to the case of uniform approximations by complex polynomials, there are no topological criteria for the existence of such approximations. We obtain a criterion for the existence ofn-analytic polynomial approximations in terms of analytic properties of these curves.

Let $E$ be a compact subset of the unit interval $[0,1]$, and let $C(E)$ denote the space of functions continuous on $E$ with the uniform norm. Consider the densely defined operator $D:C(E) \to C(E)$ given by $Dp = p’$ for all polynomials $p$. Let $G$ represent the graph of $D$, that is $G = \{ (p,p’):p$ polynomials} considered as a submanifold of $C(E) \times C(E)$. Write the interior of the set $E,\;\operatorname {int} E$ as a countable union of disjoint open intervals and let $\widehat {E}$ be the union of the closure of these intervals. The main result is that the closure of $G$ is equal to the set of all functions $(h,k) \in C(E) \times C(E)$ such that $h$ is absolutely continuous on $\widehat {E}$ and $k|\widehat {E} = h’|\widehat {E}$. As a consequence, the operator $D$ is closable if and only if the set $E$ is the closure of its interior. On the other extreme, $G$ is dense in $C(E) \times C(E)$ i.e. for any pair $(f,g) \in C(E) \times C(E)$, there exists a sequence of polynomials $\{ {p_n}\}$ so that ${p_n} \to f$ and ${p’_n} \to g$ uniformly on $E$, if and only if the interior $\operatorname {int} E$ of $E$ is empty.

In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for a class of semi-linear elliptic partial differential equations, in the context of linear iterations and non-matching grids. More precisely, making use of the subsolution-based concept, we prove that finite element Schwarz iterations converge, in the maximum norm, to the true solution of the PDE. We also give numerical results to validate the theory. This work introduces a new approach and generalizes the one in [14] as it encompasses a larger class of problems.

We study an approximate method for solving singular integral equations. It implies an approximation of a singular operator by means of a compound quadrature formula similar to the rectangle one. The corresponding systems of linear algebraic equations are solvable if so is the integral equation, while its coefficients satisfy the strong ellipticity condition. Under these restrictions we obtain a bound for the rate of convergence of solutions of systems of linear equations to the solution of the considered integral equation in the uniform vector norm.

The paper concerns the uniform polynomial approximation of a function f, continuous on the unit Euclidean sphere of ℝ3 and known only at a finite number of points that are somehow uniformly distributed on the sphere. First, we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from n − m up to n + m, being m = θn for any fixed parameter 0 < θ < 1. As n tends to infinity, we prove that these polynomials uniformly converge to f at the near-best polynomial approximation rate. Moreover, for fixed n, by using the same data points, we can further improve the approximation by suitably modulating the action ray m determined by the parameter θ. Some numerical experiments are given to illustrate the theoretical results.

An improved uniformly convergent scheme in space for 1D parabolic reaction–diffusion systems

In this work, we examine the approximation capabilities of deep neural networks utilizing the Rectified Quadratic Unit (ReQU) activation function, defined as max(0,x)2, for approximating Hölder-regular functions with respect to the uniform norm. We constructively prove that deep neural networks with ReQU activation can approximate any function within the R-ball of r-Hölder-regular functions (Hr,R([-1,1]d)) up to any accuracy ϵ with at most Oϵ-d/2r neurons and fixed number of layers. This result highlights that the effectiveness of the approximation depends significantly on the smoothness of the target function and the characteristics of the ReQU activation function. Our proof is based on approximating local Taylor expansions with deep ReQU neural networks, demonstrating their ability to capture the behavior of Hölder-regular functions effectively. Furthermore, the results can be straightforwardly generalized to any Rectified Power Unit (RePU) activation function of the form max(0,x)p for p≥2, indicating the broader applicability of our findings within this family of activations.