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
In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the c...In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.展开更多
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
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 Natural Science Foundation of Inner Mongolia(Grant No.2011MS0103)supported by National Natural Science Foundation of China(Grant No.10671019)
文摘In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.
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).
