Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I

We obtain order estimates for linear widths of the Besov classes of periodic functions of many variables in the space L _q for certain values of the parameters p and q.     

周期Besov函数类的N-宽度

本文研究了带有标准信息的各向同性Besov周期函数类的逼近问题,求得了带混淆范数的多维周期Besov函数类的Kolmogorov,Gel'fand和线性N-宽度的精确阶．     

Estimates for Approximation Characteristics of the Besov Classes $$B_{p,\theta }^r $$ of Periodic Functions of Many Variables in the Space Lq. II

We obtain order estimates for the approximation of the classes of periodic functions of many variables in the space L _q by using operators of orthogonal projection and linear operators satisfying certain conditions.     

Estimation of Some Approximating Characteristics of the Classes of Periodic Functions of One and Many Variables

Ukrainian Mathematical Journal - We establish the exact-order estimates for some approximating characteristics of the classes $$ {mathbbm{W}}_{p,alpha}^r $$ and $$ {mathbbm{B}}_{p,theta}^r $$...     

N-Widths and Average Widths of Besov Classes in Sobolev Spaces

In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.     

Approximation of Classes of Periodic Functions in Several Variables

We study the approximation of the classes and of periodic functions of several variables by multiple Fourier sums of fixed order constructed with regard to individual properties of functions from these classes. In a number of cases, such approximations allow us to achieve a better degree of approximation of the classes indicated above as compared to their approximation by staircase hyperbolic Fourier sums.     

各向异性Sobolev及Besov类的平均单边宽度

给出了各向异性Sobolev类及各向异性Besov类的平均单边宽度,获得了相应的弱渐近估计.     

The Infinite-Dimensional Widths and Optimal Recovery of Generalized Besov Classes

The purpose of the present paper is to consider some weak asymptotic problems concerning the infinite-dimensional Kolmogorov widths, the infinite-dimensional linear widths, the infinite-dimensional Gel'fand widths and optimal recovery in Besov space. It is obvious to be found that the results obtained and methods used in Besov space are easily generalized, and hence in this paper these results in an extension of Besov spaces on R^d are stated and proved. Meantime, the representation theorem and approximation of these spaces by polynomial splines are discussed.     

The Average Widths of Sobolev-Wiener Classes and Besov-Wiener Classes

This paper concerns the problem of average σ-width of Sobolev-Wiener classes W^rpq(R^d),W^rpq(M,R^d),and Besov-Wiener classes S^rpqθb(R^d).S^rpqθB(R^d),S^rpqθb(M,R^d),S^rpqθB(R^d)in the metric Lq(R^d) for 1≤q≤p≤∞.The weak asymptotic results concerning the average linear widths,the average Bernstein widths and the infinite-dimensional Gel‘fand widths are obtained,respectively.     

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ _β ⁻ whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 946–971, July, 2005.     

Approximation of Classes of $${\bar \psi }$$ -Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series

We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of -integrals of functions of many variables. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 564–570, April, 2005.     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. II

In normed spaces of functions analytic in the Jordan domain , we establish exact order estimates for the Kolmogorov widths of classes of functions that can be represented in by Cauchy-type integrals along = with densities f(·) such that . Here, is a conformal mapping of onto {w: |w| > 1}, and is a certain subset of infinitely differentiable functions on T = {w: |w| = 1}.     

Approximation of Periodic Functions of Many Variables by Functions of a Smaller Number of Variables in Orlicz Metric Spaces

Ukrainian Mathematical Journal - For periodic functions of many variables, we propose a method for their approximation in the Orlicz spaces L(??m). According to this method, the...