最坏框架与平均框架下区间[1,1]上带Jacobi权的函数逼近

本文研究最坏框架和平均框架下区间[1,1]上带Jocobi权(1 x)α(1+x)β,α,β1/2的函数逼近问题.在最坏框架下,本文得到加权Sobolev空间BWr p,α,β在Lq,α,β(1 q∞)空间尺度下的Kolmogorov n-宽度和线性n-宽度的渐近最优阶,其中Lq,α,β(1 q∞)表示区间[1,1]上带Jacobi权的加权Lq空间.在平均框架下,本文研究具有Gauss测度的加权Sobolev空间Wr2,α,β被多项式子空间和Fourier部分和算子在Lq,α,β(1 q∞)空间尺度下的最佳逼近问题,得到平均误差估计的渐近阶.我们发现,在平均框架下,多项式子空间和Fourier部分和算子在Lq,α,β(1 q2+22 max{α,β}+1)空间尺度下是渐近最优的线性子空间和渐近最优的线性算子.

Approximation of functions on [-1, 1] with Jacobi weights in the worst and average case settings

We study the weighted approximation of functions on the interval [-1, 1] with Jocobi weights （1 -x）α（1 ＋ x）β, α, β〉 -1/2 in the worst and average case settings. In the worst case setting, we discuss the Kolmogorov n-widths dn（BW^r p,α,β） and linear n-widths 5n（BW^r p,α,β） of the weighted Sobolev classes BW^r p,α,β on [-1, 1], where BW^r p,α,β 1≤q〈∞, denotes the Lq space on [-1, 1] with respect to Jocobi weights. Optimal asymptotic orders of dn（BW^r p,α,β,Lq p,α,β） and 5n（BW^r p,α,β,Lq p,α,β） as n → ∞ are obtained for all 1 K p, q K ∞. In the average case setting, we investigate the best approximation of functions on the weighted Sobolev class BWq α,β equipped with a centered Gaussian measure by polynomial subspaces in the Lq,α,β metric for Lq α,β. The asymptotic orders of the average error estimations are obtained. It turns out that in the average case setting, the polynomial subspaces are the asymptotically optimal subspaces in the Lq,α,β metric only for 1 〈 q 〈 2 ＋ 2/（2max{α,β} ＋ 1）.