Bernstein n-widths for classes of convolution functions with kernels satisfying certain oscillation properties

In this paper, we consider some classes of 2π-periodic convolution functions Bp, and Kp with kernels having certain oscillation properties, which include the classical Sobolev class as special case. With the help of the spectral of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes Bp, Kp in the space Lp for 1 〈 p 〈 ∞.     

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.     

Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and general-ized Sobolev classes into Lq([0,1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0,1]d)) to Lq([0,1]d) space for all 1 q,p ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.     

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.     

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.     

AVERAGE KOLMOGOROV N-WIDTHS (N-K WIDTH) AND OPTIMAL INTERPOLATION OF SOBOLEV CLASS IN L_p(R)

The author obtains the exact values of the average n-K widths for some Sobolev classes defined by an ordinary differential operator P(D)=multiply from i=1 to r(D-t_il), t_i∈R, in the metric L_(R), 1≤p≤∞, and identifies some optimal subspaces. Furthermore, the optimal interpolation problem for these Sobolev classes is considered by sampling the function values at some countable sets of points distributed reasonably on R, and some exact results are obtained.     

Approximation of functions on the Sobolev space with a Gaussian measure

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Valle-Poussin operators, Ces`aro operators, Abel opera-tors, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the L q space for 1≤q ∞, and the Fourier partial summation operators and the Valle-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the L q space for 1≤q ∞.     

COMPUTATIONAL COMPLEXITY IN WORST,STOCHASTIC AND AVERAGE CASE SETTING ON FUNCTIONAL APPROXIMATION PROBLEM OF MULTIVARIATE

The order of computational complexity of all bounded linear functional ap proximation problem is determined for the generalized Sobolev class W_p~(?)(Id), Nikolskii class H|∞~k(Id) in the worst (deterministic), stochastic and average case setting, from which it is concluded that the bounded linear functional approximation problem for the classes W_p~(?)(Id) and H_∞~k(Id) is intractable in worst case setting, but is tractable with respect to stochastic and average case setting.     

PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T)(1≤Q≤∞)

Probabilistic linear(N,δ)-widths and p-average linear N-widths of Sobolev space W_2~τ(T),equipped with a Gaussian probability measure μ,are studied in the metric of S_q(T)(1 q oo),and determined the asymptotic equalities:and where 0p∞,δ∈(0,1/2],ρl,and Sq(T) is a subspace of L_1(T),in which the Fourier series is absolutely convergent in l_q sense.     

AVERAGE ONESIDED WIDTHS OF SOBOLEV AND BESOV CLASSES

The article concerns the average onesided widths of the Sobolev and Besov classes and the classes of functions with bounded moduli of smoothness. The weak asymptotic results are obtained for the corresponding quantities.     

Ismagilov Type Theorems for Linear, Gel′fand and Bernstein n-Widths

Using a variational principle for s-numbers, we obtain estimates for the linear, Gel′fand. and Bernstein n-widths. A simple proof of some results concerned with the exact values of n-widths of diagonal operators is given. We also calculate the exact values at the Bernstein n-widths for the Hardy-Sobolev classes.     

On the widths of Sobolev classes

Optimal estimates of Kolmogorov’s n-widths, linear n-widths and Gelfand’s n-widths of the weighted Sobolev classes on the unit sphere S^d are established. Similar results are also established on the unit ball B^d and on the simplex T^d.     

Best Linear Approximation Methods for Some Classes of Analytic Functions on the Unit Disk

Considering Banach Hardy spaces and weighted Bergman spaces, we find the sharp values of the Bernstein, Kolmogorov, Gelfand, and linear n-widths for the classes of analytic functions on the unit disk whose moduli of continuity of the rth derivatives averaged with weight are majorized by a given function satisfying some constraints.

     

The Average Widths and Non-linear Widths of the Classes of Multivariate Functions with Bounded Moduli of Smoothness

The classes of the multivariate functions with bounded moduli on Rd and Td are given and their average σ-widths and non-linear n-widths are discussed. The weak asymptotic behaviors are established for the corresponding quantities.     

多元Sobolev关于L_(qp)(R~d)下的平均线性宽度

得到了两种多元 Sobloev类于 Lqp( Rd )下平均线性宽度的弱渐进估计 .     

用指数型整函数的最佳限制逼近

我们研究了由仅有实零点的代数多项式导出的微分算子确定的广义Sobolev类利用指数型整函数作为逼近工具的最佳限制逼近问题.利用Fourier变换和周期化等方法,得到在L_2(R)范数下的广义Sobolev光滑函数类的相对平均宽度和最佳限制逼近的精确常数,以及当0是这个代数多项式的一个至多2重的零点时,得到最佳限制逼近在L_1(R)范数和一致范数下的广义Sobolev类的精确到阶的结果.     

The Gelfand and Bernsteinn-widths of some classes of analytic functions

This paper contains generalizations of a well-known theorem of Ismagilov on Kolmogorovn-widths in a Hilbert space for Bernstein and Gelfandn-widths. Some examples are considered. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 112–129.     

Decomposition theorems and approximation by a ``floating" system of exponentials

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.