某些周期函数类的n宽度

设G是B核,记用d_n,dⁿ和δ_n分别表示Kolmogorov,Gel''fand和线性n宽度。本文求出了和的精确值,找到了各自的极子空间(或最优算子)。由此证明了Pinkus猜想(即是的极子空间,)在p=q时的正确性。     

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

The author obtains the excat values of the average n-K widths for some Sobolev classes defined by an ordinary differential operator $\[P(D) = \prod\limits_{i = 1}^r {D - {t_i}I),{t_i} \in R} \]$,in the metric L_p(R),$1\leq p\leq \infty$,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.     

实轴上函数类WrHω的平均n─宽度

对于ｒ阶导数的连续模被一个给定上凸连续模新控制的所有ｒ阶可微函数类，我们求出在L_∞（Ｒ）一范数下其平均ｎ－宽度，并找到了极优子空间。     

实轴上函数类W~rH~ω的平均n─宽度

对于ｒ阶导数的连续模被一个给定上凸连续模新控制的所有ｒ阶可微函数类，我们求出在ｌｏｏ（Ｒ）一范数下其平均ｎ－宽度，并找到了极优子空间。     

The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.     

函数型数据支持向量回归

函数型数据回归是一个非常有意义的课题.已有工作都是利用平方损失来衡量误差,而本文采用ε-不敏感损失来衡量误差.本文构造基于ε-不敏感损失的逼近元,给出表示形式及其系数计算.逼近元具有鲁棒性和稀疏性等性质.本文的主要结果是,在一些常规条件下建立预测误差收敛阶.与关于平方损失工作相比,我们不要求协方差算子与积分算子之间的"对齐"关系.此外,本文还讨论了支持向量回归函数本身的逼近性质.即使对有限维数据,关于这方面的结果在文献中也尚未见到.     

任意细分面具的级联算法对单个初始函数的收敛性

级联算法在计算机图形和小波分析中都有很重要的作用．对任意的初始函数φ0，一个级联序列(φ_n)_(n=1)~∞是由迭代产生的序列φ_n=C_aφ_(n-1)(n=1，2，…)，其中 C_a 定义为C_ag=sum from α∈Ζa(α)g(2·-α)，g∈L_p(R)．用函数序列和联合谱半径刻画了级联序列的收敛性．作为一个结果，证明了任意的级联收敛序列都有几何收敛速度，即‖φ_(n-1)-φ_n‖_[L_p(R)]=O((?)~n)对某个(?)∈(0，1)成立．不要求对面具的求和定则的条件．     

周期变号减少核确定的完全样条与最优取样点的唯一性

证明了具有给定零点的完全样条的唯一性，证明了在卷积类上最优取样点的唯一性．     

正则化回归算法的快速学习率

本文讨论了再生核Hilbert 空间上一类广泛的正则化回归算法的学习率问题. 在分析算法的样本误差时, 我们利用了一种复加权的经验过程, 保证了方差与惩罚泛函同时被阈值控制, 从而避免了繁琐的迭代过程. 本文得到了比之前文献结果更为快速的学习率.     

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.