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1.
In this paper we investigate the
probabilistic linear $(n,\delta)$-widths and $p$-average linear
$n$-widths of the Sobolev space $W^r_2$ equipped with the Gaussian
measure $\mu$ in the $L_{\infty}$-norm, and determine the
asymptotic equalities
\begin{eqnarray*}
\lambda_{n,\delta}(W^r_2,\mu,L_{\infty})
&\asymp&\frac{\sqrt{\ln
(n/\delta)}}{n^{r+(s-1)/2}},\\[3pt]
\lambda^{(a)}_n(W^r_2,\mu,L_{\infty})_p &\asymp&\frac{\sqrt{\ln
n}}{n^{r+(s-1)/2}}, \qquad 0 < p < \infty.
\end{eqnarray*} 相似文献
2.
Presenting a unified approach, we establish a Kolmogorov-type comparison theorem for classes of 2π-periodic functions defined
by a special class of operators having certain oscillation properties, which include the classical Sobolev class of 2π-periodic
functions, the Achieser class, and the Hardy-Sobolev class as examples. Then, using these results, we prove a Taikov-type
inequality, and calculate the exact values of the Kolmogorov, Gel'fand, linear, and information n-widths of these classes
of functions in the space Lq, which is the classical Lebesgue integral space of 2π-periodic functions with the usual norm. 相似文献
3.
First we study several extremal problems on minimax, and prove that they are equivalent. Then we connect this result with the exact values of some approximation characteristics of diagonal operators in different settings, such as the best n-term approximation, the linear average and stochastic n-widths, and the Kolmogorov and linear n-widths. Most of these exact values were known before, but in terms of equivalence of these extremal problems, we present a unified approach to give them a direct proof. 相似文献
4.
Gensun Fang 《中国科学A辑(英文版)》1998,41(12):1272-1277
The Nikolskii type inequality for cardinal splines
is proved, which is exact in the sense of order, where ∈ ℒ
m,h
, and ℒ
m,k
is the space of cardinal splines with nodes
Project supported by the National Natural Science Foundation of China (Grant No. 19671012), and Doctoral Programme Foundation
of Institution of Higher Education. 相似文献
5.
OPTIMALQUADRATUREOFTHESOBOLEVCLASSW_1~r(R)DEFINEDONWHOLEREALAXIS(房艮孙,刘永平)¥FangGensun;LiuYongping(Dept.ofMath.,BeijingNormalUni... 相似文献
6.
In this paper, we obtain the strong assymptotic values of n-width dn(WrHω, C)and dn(WrHω, L) for non-periodic function class WrHω, wheve ω is a concave modulus of continuity. 相似文献
7.
Hermite型多元样本定理及Sobolev类上混淆误差的估计 总被引:1,自引:0,他引:1
本文证明了Hermite型多元样本定理,并由此确定了Sobolev类上混淆误差阶的精确估计. 相似文献
8.
We consider some classes of 2π-periodic functions defined by a class of operators having certain oscillation properties, which include the classical Sobolev class and a class of analytic functions which can not be represented as a convolution class as its special cases. Let be the largest integer not bigger than x. We prove that on these classes of functions the rectangular formula
is optimal among all quadrature formulae of the form
where the nodes 0 ≤ t
1 < ... < t
n
< 2π and the coefficients (weights) are arbitrary, i = 1,...,n, j = 0,1,..., ν
i
− 1, and (ν1,...,ν
n
) is a system of positive integers satisfying the condition . In particular, the rectangular formula is optimal for these classes of functions among all quadrature formulae of the form
with free nodes 0 ≤ t
1 < ... < t
N
< 2π and arbitrary weights . Moreover, we exactly determine the error estimates of the optimal quadrature formulae on these classes of functions.Project supported by the National Natural Science Foundation of China (Grant No. 10671019) and Research Fund for the Doctoral Program Higher Education (Grant No. 20050027007). 相似文献
9.
In this paper, we determine the asymptotic values of the probabilistic adaptive widths of the space of multivariate functions
with bounded mixed derivative (MW2r(Td),μ) relative to the manifold (YN,ν) in the Lq(Td)-norm, 1 < q ≤ 2, where μ and ν are two given Gaussian measures. 相似文献
10.
In this paper we study the Gelfand and Kolmogorov numbers of Sobolev embeddings between weighted function spaces of Besov and Triebel–Lizorkin type with polynomial weights. The sharp asymptotic estimates are determined in the so-called non-limiting case. 相似文献