Theorems Of Approximation Theory Research Articles

Application of some fixed-point theorems in approximation theory

In this paper, we study some fixed point theorems in approximation theory. Fixed point theory initially emerged in the article demonstrating existence of solutions of differential equations .which appeared in the second quarter of the 18th century. Lateral on this technique was improved as a method of successive approximations which was extracted and abstracted as a fixed point theorem in the framework of complete normed space. It is stated that fixed point theory is initiated by Stefan Banach. Fixed point theorem have been used in many instances in approximation theory. Brosowski gave some application of fixed point theory for approximation.

Approximation by Rational Functions in Variable Exponent Morrey–Smirnov Classes

In this work, the direct theorem of approximation theory in variable exponent Morrey–Smirnov classes of analytic functions, defined on a doubly connected domain of the complex plane bounded by two sufficiently smooth curves, is investigated.

Smoothness of functions and Fourier coefficients

We consider functions represented as trigonometric series with general monotone Fourier coefficients. The main result of the paper is the equivalence of the modulus of smoothness, , of such functions to certain sums of their Fourier coefficients. As applications, for such functions we give a description of the norm in the Besov space and sharp direct and inverse theorems in approximation theory. Bibliography: 34 titles.

A direct theorem of approximation theory for a general modulus of smoothness

We introduce the notion of general modulus of smoothness in the spaces Lp of 2π-periodic pth-power integrable functions; in these spaces, the coefficients multiplying the values of a given function at the nodes of the uniform lattice are the Fourier coefficients of some 2π-periodic function called the generator of the modulus. It is shown that all known moduli of smoothness are special cases of this general construction. For the introduced modulus, in the case 1 ≤ p ≤ +∞ we prove a direct theorem of approximation theory (a Jackson-type estimate). It is shown that the known Jackson-type estimates for the classical moduli, the modulus of positive fractional order, and the modulus of smoothness related to the Riesz derivative are its direct consequences. We also obtain a universal structural description of classes of functions whose best approximations have a certain order of convergence to zero.

Uniform approximation with radical functions

In this work we give a family of radical functions, C, uniformly dense in C [a, b]. This family does not satisfy the usual theorems of Approximation Theory about C [a, b]. The mathematical expression of the sequence that converges uniformly to every ƒ ∈ C[a, b] is determined. We present algorithms to approximate ƒ with the uniform norm in different practical instances. A linear subspace of radical functions that verifies the Haar condition is constructed. We also study the degree of uniform approximation by radical functions and we show a comparative analysis with the approximation by arbitrary polynomials and Bernstein polynomials.

Approximation of Functions by Rational Functions on Closed Curves of the Complex Plane

Let Γ be a quasi-smooth curve in the complex plane \({\mathbb{C}}\) . In this study, a direct theorem of approximation theory in certain subclasses of the functions which have continuous derivatives through order r on a closed curve Γ is proved.

Direct and inverse theorems in approximation theory for periodic functions in S.B. Stechkin’s papers and the development of these theorems

S.B. Stechkin made a considerable contribution to the important aspect of approximation theory for periodic functions concerned with direct and inverse theorems. We present the history of developing this research area and pose new problems.

New theorems in approximation theory

The aim of this paper is to prove some results for equivalence of moduli of smoothnes in approximation theory , we used a"non uniform" modulus of smoothness and the weighted Ditzian –Totik moduli of smoothness in by spline functions ,several results are obtained .For example , it shown that ,for any the inequality , is satisfied ,finally, similar result for chebyshev partition and weighted Ditzian –Totik moduli of smoothness are also obtained.

New theorems in approximation theory

The aim of this paper is to prove some results for equivalence of moduli of smoothnes in approximation theory , we used a"non uniform" modulus of smoothness and the weighted Ditzian –Totik moduli of smoothness in by spline functions ,several results are obtained .For example , it shown that ,for any the inequality , is satisfied ,finally, similar result for chebyshev partition and weighted Ditzian –Totik moduli of smoothness are also obtained.

On the connection between the best approximation by algebraic polynomials and the modulus of smoothness of order r

We introduce a family of asymmetric generalized translations. In terms of these operators, we define generalized moduli of smoothness for which the direct and inverse theorems of approximation theory are proved.

Invariance Theorems in Approximation Theory and their Applications

Let B be a closed linear subspace of a Banach space F and let \(\{T_s\}_{s\in G}\) be a group of continuous linear operators \(T_s:F\to F\), where G is a compact topological group. We prove that if \(f\in F\) is invariant under \(\{T_s\}_{s\in G}\), then under some conditions on f, F, B, and G, there exists an element \(g^*\in B\) of best approximation to f that has the same property. As applications, we compute the bivariate Bernstein constant for \(L_1\) polynomial approximation of \(|x|^{\lambda}\) and solve a Braess problem on the exponential order of decay of the error of polynomial approximation of \(|x-a|^{-{\lambda}}\). Other examples and applications are discussed as well.

Refined theorems of approximation theory in the space of p-absolutely continuous functions

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in Lp generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp.

Short proofs of some theorems in approximation theory

New proofs, shorter than previous ones, of three theorems from approximation theory are given. The Schwarz's lemma is a key tool.

Negative Theorems in Approximation Theory

Negative Theorems in Approximation Theory

Negative Theorems in Approximation Theory

Negative Theorems in Approximation Theory

Properties of a Family of Operators

In this paper we define a family of new nonsymmetric operators of generalized translation and describe their properties. For each of these operators, we introduce a generalized modulus of smoothness, for which the direct and inverse theorems of approximation theory are given.

An inverse theorem of approximation theory of periodic functions in various metrics

An inverse theorem of approximation theory of periodic functions in various metrics

Limit theorems in approximation theory

В РАБОтЕ ОпРЕДЕлЕНы У слОВИь, ОБЕспЕЧИВАУЩ ИЕ ВыпОлНЕНИЕ сООтНОшЕ НИИ ВИДА $$\begin{array}{*{20}c} {\mathop {\lim }\limits_{n \to \infty } E(h_n ,B_n ,F_n ) = E(f,B,F),} \\ {\mathop {\lim }\limits_{n \to \infty } \mathop {\sup }\limits_{h_n \in K_n } E(h_n ,B_n ,F_n ) = \mathop {\sup }\limits_{f \in K} E(f,B,F),} \\ \end{array}$$ гДЕE(g, H, M) — НАИлУЧшЕЕ пРИ БлИжЕНИЕ ЁлЕМЕНтАg ЁлЕМЕНтАМИ ИжH В МЕтР ИкЕ НОРМИРОВАННОгО пРОстРАНстВАM, Kn,K — НЕк ОтОРыЕ МНОжЕстВА ЁлЕ МЕНтОВ.

Order Stars, Approximations and Finite Differences II. Theorems in Approximation Theory

The theory of order stars is applied to two problems in approximation theory: determination of upper bounds on the size of blocks in the Pade tableau of an analytic function and the task of highest-order approximation of a stable analytic function by a contraction. It is shown that in both cases global properties of the underlying function—the loci and the nature of essential singularities, zeros and poles—lead to realistic upper bounds. The given theory is applied to a range of examples that illustrate the potential of this approach.

CONSTRUCTION AND INVESTIGATION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS BY METHODS IN THE THEORY OF APPROXIMATION OF FUNCTIONS

The steady-state equation , the parabolic Cauchy problem , , and the hyperbolic problem , , , are considered, where is a matrix-valued positive selfadjoint second-order partial differential operator with analytic coefficients, and is an analytic function. Methods in the theory of weighted approximation of functions by polynomials on the line are used to construct polynomial representations of solutions of these problems of the form , where the polynomials , , are constructed in explicit form. Estimates of the rate of convergence are given. With the help of these estimates and Bernstein's inverse theorems in approximation theory, theorems are obtained on the smoothness and analyticity of solutions of degenerate systems whose coefficients are trigonometric polynomials.Bibliography: 9 titles.