In this paper, an extension of Besov classes of periodic functions on Td is given. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, and the Gel＇fand n-widths are obtained, respecti...In this paper, an extension of Besov classes of periodic functions on Td is given. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, and the Gel＇fand n-widths are obtained, respectively.展开更多
In this paper we will show that if an approximation process {Ln}n∈N is shape- preserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1], and the operators Ln a...In this paper we will show that if an approximation process {Ln}n∈N is shape- preserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1], and the operators Ln are assumed to be of finite rank n, then the order of convergence of D^kLnf to D^kf cannot be better than n-2 even for the functions x^k, x^k＋1, x^k＋2 on any subset of [0,1 ] with positive measure. Taking into account this fact, we will be able to find some asymptotic estimates of linear relative n-width of sets of differentiable functions in the space LP[0, 1], p ∈ N.展开更多
Let F(x):R^m→R^m be an odd,continuously differentiable homogeneous map.The pa- per is devoted to the critical points of the generalized Rayleigh ratio||F(x)||_(l_q^m)/||x||_(l_p^m)and connected with some problems of ...Let F(x):R^m→R^m be an odd,continuously differentiable homogeneous map.The pa- per is devoted to the critical points of the generalized Rayleigh ratio||F(x)||_(l_q^m)/||x||_(l_p^m)and connected with some problems of the approximation theory.We find the lower bound for Kolmogorov n-width d_n(F(Bl_p^m),l_q^m).展开更多
Given a separable real Banach space F and a separable real Hilbert space X, we denote by S a bounded linear operator from F into X. Let μ be a Gaussian measure defined on the Borel field of F with mean element m_μ= ...Given a separable real Banach space F and a separable real Hilbert space X, we denote by S a bounded linear operator from F into X. Let μ be a Gaussian measure defined on the Borel field of F with mean element m_μ= 0 and correlation operator C_μ. By [1], C_μ is a bounded linear operator from F~*→F such展开更多
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 ...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 〈 ∞.展开更多
This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr（D） = DσПlj=1 （D2 - tj2I） in Orlicz spaces. The relationship between the extreme value problem and width theory is ...This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr（D） = DσПlj=1 （D2 - tj2I） in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov＇s widths, Gelfand＇s widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given.展开更多
In this paper,we consider the relative n-widths of two kinds of periodic convolution classes,Kp(K) and Bp(G),whose convolution kernels are NCVD-kernel K and B-kernel G. The asymptotic estimations of Kn(Kp(K),Kp(K))q a...In this paper,we consider the relative n-widths of two kinds of periodic convolution classes,Kp(K) and Bp(G),whose convolution kernels are NCVD-kernel K and B-kernel G. The asymptotic estimations of Kn(Kp(K),Kp(K))q and Kn(Bp(G),Bp(G))q are obtained for p=1 and ∞,1≤ q≤∞.展开更多
Let <sub>p</sub> = {a +G*h|a∈R, h⊥1, ||h||<sub>p</sub>≤1}, where G is a B-kernel. We obtain the exactvalues of the d<sub>2n</sub>（B<sub>p</sub>, L<sub>p</su...Let <sub>p</sub> = {a +G*h|a∈R, h⊥1, ||h||<sub>p</sub>≤1}, where G is a B-kernel. We obtain the exactvalues of the d<sub>2n</sub>（B<sub>p</sub>, L<sub>p</sub>）, d<sup>2n</sup>（B<sub>p</sub>,L<sub>p</sub>） and δ<sub>2n</sub>（B<sub>p</sub>,L<sub>p</sub>） for p∈（1,+∞）/{2}. Furthermore. weidentify some optimal subspaces for d<sub>2n</sub> and d<sup>2n</sup> respectively, and construct an optimallinear operator of rank 2n for δ<sub>2n</sub>, from which we answer affirmatively Pinkus Conjecture,i.e. ?p≥q≥1, W<sub>2n</sub>={a +sum from i=0 to （2n-1） a<sub>i</sub>G（x-iπ/n）a,a<sub>i</sub>∈R,sum from i=0 to（2n-1） a<sub>i</sub>=0} is an optimal subspace ford<sub>2n</sub>（?<sub>p</sub>,L<sub>p</sub>） for p=q.展开更多
1 Introduction Let Q（x）be a polynomial with real coefficients and the Sobolev class W<sub>p</sub>（Q（D））be the set of continuous 2π-periodic functions f（x） for which f<sup>（degQ-1）</sup&...1 Introduction Let Q（x）be a polynomial with real coefficients and the Sobolev class W<sub>p</sub>（Q（D））be the set of continuous 2π-periodic functions f（x） for which f<sup>（degQ-1）</sup> is absolutely continuous and ‖Q（D）f‖<sub>p</sub>≤1, where degQ is the degree of Q, D=d/dt, and ‖·‖<sub>p</sub> is the usual L<sub>p</sub>[0, 2π]-展开更多
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 ...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.展开更多
文摘In this paper, an extension of Besov classes of periodic functions on Td is given. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, and the Gel＇fand n-widths are obtained, respectively.
基金Supported by RFBR(grant10-01-00270)the president of the Russian Federation(NS-4383.2010.1)
文摘In this paper we will show that if an approximation process {Ln}n∈N is shape- preserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1], and the operators Ln are assumed to be of finite rank n, then the order of convergence of D^kLnf to D^kf cannot be better than n-2 even for the functions x^k, x^k＋1, x^k＋2 on any subset of [0,1 ] with positive measure. Taking into account this fact, we will be able to find some asymptotic estimates of linear relative n-width of sets of differentiable functions in the space LP[0, 1], p ∈ N.
文摘Let F(x):R^m→R^m be an odd,continuously differentiable homogeneous map.The pa- per is devoted to the critical points of the generalized Rayleigh ratio||F(x)||_(l_q^m)/||x||_(l_p^m)and connected with some problems of the approximation theory.We find the lower bound for Kolmogorov n-width d_n(F(Bl_p^m),l_q^m).
基金Project supported by the National Natural Science Foundation of China.
文摘Given a separable real Banach space F and a separable real Hilbert space X, we denote by S a bounded linear operator from F into X. Let μ be a Gaussian measure defined on the Borel field of F with mean element m_μ= 0 and correlation operator C_μ. By [1], C_μ is a bounded linear operator from F~*→F such
基金supported by the Natural Science Foundation of China (Grant No. 10671019)Research Fund for the Doctoral Program Higher Education (No. 20050027007)Scientific Research Fund of Zhejiang Provincial Education Department (No. 20070509)
文摘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 〈 ∞.
基金Supported by National Natural Science Foundation of China(Grant No.11161033)
文摘This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr（D） = DσПlj=1 （D2 - tj2I） in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov＇s widths, Gelfand＇s widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given.
基金supported by National Natural Science Foundation of China (Grant No.10771016)the "985" Programme of Beijing Normal University
文摘In this paper,we consider the relative n-widths of two kinds of periodic convolution classes,Kp(K) and Bp(G),whose convolution kernels are NCVD-kernel K and B-kernel G. The asymptotic estimations of Kn(Kp(K),Kp(K))q and Kn(Bp(G),Bp(G))q are obtained for p=1 and ∞,1≤ q≤∞.
基金Project supported by the National Natural Science Foundation of China.
文摘Let <sub>p</sub> = {a +G*h|a∈R, h⊥1, ||h||<sub>p</sub>≤1}, where G is a B-kernel. We obtain the exactvalues of the d<sub>2n</sub>（B<sub>p</sub>, L<sub>p</sub>）, d<sup>2n</sup>（B<sub>p</sub>,L<sub>p</sub>） and δ<sub>2n</sub>（B<sub>p</sub>,L<sub>p</sub>） for p∈（1,+∞）/{2}. Furthermore. weidentify some optimal subspaces for d<sub>2n</sub> and d<sup>2n</sup> respectively, and construct an optimallinear operator of rank 2n for δ<sub>2n</sub>, from which we answer affirmatively Pinkus Conjecture,i.e. ?p≥q≥1, W<sub>2n</sub>={a +sum from i=0 to （2n-1） a<sub>i</sub>G（x-iπ/n）a,a<sub>i</sub>∈R,sum from i=0 to（2n-1） a<sub>i</sub>=0} is an optimal subspace ford<sub>2n</sub>（?<sub>p</sub>,L<sub>p</sub>） for p=q.
基金Project supported by the National Natural Science Foundation of China.
文摘1 Introduction Let Q（x）be a polynomial with real coefficients and the Sobolev class W<sub>p</sub>（Q（D））be the set of continuous 2π-periodic functions f（x） for which f<sup>（degQ-1）</sup> is absolutely continuous and ‖Q（D）f‖<sub>p</sub>≤1, where degQ is the degree of Q, D=d/dt, and ‖·‖<sub>p</sub> is the usual L<sub>p</sub>[0, 2π]-
基金Supported by Project(No.10471010)of National Natural Science Foundation of ChinaSupported by the Development Foundation of Science and Technology of Tianjin Universities(20040405)Supported by Project"Representation Theory and Related Topics"of the"985 Program"of Beijing Normal University
文摘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.
