Approximation of classes of periodic functions of several variables by nuclear operators

Translated from Matematicheskii Zametki, Vol. 47, No. 3, pp. 32–41, March, 1990.     

Approximation of functions of two variables by certain linear positive operators

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an rth order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.     

Approximation in Banach space by linear positive operators

In this paper, we obtain a sufficient condition for the convergence of positive linear operators in Banach function spaces on \({\mathbb {R}}^n\) and derive a Korovkin type theorem for these spaces. Also, we generalized this result via statistical sense.     

Estimates for widths of classes of periodic multivariable functions

Doklady Mathematics -     

Approximation with arbitrary order by certain linear positive operators

This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose by using a strictly decreasing positive sequence \((\lambda _n)_{n\ge 1}\) such that \(\lim \limits _{n\rightarrow \infty }\lambda _n=0\), how fast we want. Particular cases are presented.     

Approximation theorems by positive linear operators in weighted spaces

In this study, we obtain some Korovkin type approximation theorems by positive linear operators on the weighted space of all real valued functions defined on the real two-dimensional Euclidean space \mathbbR²{\mathbb{R}^2}. This paper is mainly consisted of two parts: a Korovkin type approximation theorem via the concept of A-statistical convergence and a Korovkin type approximation theorem via A{\mathcal {A}}-summability.     

Approximation of classes of convolutions by linear operators of special form

A parametric family of operators G _ρ is constructed for the class of convolutions W _p,m (K) whose kernel K was generated by the moment sequence. We obtain a formula for evaluating $E(W_{p,m} (K);G_\rho )_p : = \mathop {\sup }\limits_{f \in W_{p,m} (K)} \left\| {f - G_\rho (f)} \right\|_p .$ . For the case in which W _p,m (K)=W _r,β ^p,m , we obtain an expansion in powers of the parameter ?=?ln ρ for E(W _p,m ^r,β ; G _ρ,r ) _p , where β ∈ ?, γ > 0, and m ∈ ?, while p = 1 or p = ∞.     

Approximation of classes of periodic functions of several variables

We study classes of periodic functions of several variables with bounded generalized derivative in the metric of the space L_p. We obtain order estimates of deviations of Fourier sums, which are constructed depending on the behavior of functions that define the operator of generalized differentiation. We find estimates of the Kolmogorov widths, which are realized by the Fourier sums that are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 662–672, May, 1992.     

Approximation of analytical functions by sequences of k-positive linear operators

Approximation properties of sequences of k-positive operators, i.e. linear operators acting in the space of analytical functions and preserving the cone of functions with non-negative Taylor coefficients, are studied. Some general theorems which are valid in the space of functions that are analytical in a bounded simply connected domain are proved.     

Approximation of classes of periodic functions with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.     

Unbounded functions and positive linear operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.     

Approximation properties of a generalization of positive linear operators

In the present paper we introduce a generalization of positive linear operators and obtain its Korovkin type approximation properties. The rates of convergence of this generalization is also obtained by means of modulus of continuity and Lipschitz type maximal functions. The second purpose of this paper is to obtain weighted approximation properties for the generalization of positive linear operators defined in this paper. Also we obtain a differential equation so that the second moment of our operators is a particular solution of it. Lastly, some Voronovskaja type asymptotic formulas are obtained for Meyer-König and Zeller type and Bleimann, Butzer and Hahn type operators.     

Approximation of some classes of periodic functions of many variables

We obtain the exact order of deviations of Fejér sums on the class of continuous functions. This order is determined by a given majorant of the best approximations.