首页 / 资料库 / 文献详情

最坏框架与平均框架下区间[-1, 1]上带Jacobi权的函数逼近

Xuebo Zhai、Xiuyan Hu

2014Scientia Sinica MathematicaMathematics被引 2

摘要

We study the weighted approximation of functions on the interval [-1,1] with Jocobi weights (1-<italic>x</italic>)^{<italic>α</italic>}(1+<italic>x</italic>)^{<italic>β</italic>}, <italic>α</italic>,<italic>β</italic> &gt; -1/2 in the worst and average case settings. In the worst case setting, we discuss the Kolmogorov <italic>n</italic>-widths <italic>d</italic>_{<italic>n</italic>}(<italic>BW</italic>_{<italic>p,α,β</italic>}^{<italic>r</italic>},<italic>L</italic>_{<italic>q,α,β</italic>}) and linear <italic>n</italic>-widths <italic>δ</italic>_{<italic>n</italic>}(<italic>BW</italic>_{<italic>p,α,β</italic>}^{<italic>r</italic>},<italic>L</italic>_{<italic>q,α,β</italic>}) of the weighted Sobolev classes <italic>BW</italic>_{<italic>p,α,β</italic>} on [-1,1], where <italic>L</italic>_{<italic>q,α,β</italic>},1 ≤ <italic>q</italic>≤∞ denotes the <italic>L</italic>_{<italic>q</italic>} space on [-1,1] with respect to Jocobi weights. Optimal asymptotic orders of <italic>d</italic>_{<italic>n</italic>}(<italic>BW</italic>_{<italic>p,α,β</italic>}^{<italic>r</italic>},<italic>L</italic>_{<italic>q,α,β</italic>}) and <italic>δ</italic>_{<italic>n</italic>}(<italic>BW</italic>_{<italic>p,α,β</italic>}^{<italic>r</italic>},<italic>L</italic>_{<italic>q,α,β</italic>}) as <italic>n</italic>→∞ are obtained for all 1 ≤ <italic>p,q</italic> ≤ ∞. In the average case setting, we investigate the best approximation of functions on the weighted Sobolev class <italic>B</italic><italic>W</italic>_{2,<italic>α,β</italic>}^{<italic>r</italic>} equipped with a centered Gaussian measure by polynomial subspaces in the <italic>L</italic>_{<italic>q,α,β</italic>} metric for 1 ≤ <italic>q</italic> &lt; ∞. The asymptotic orders of the average error estimations are obtained. It turns out that in the ave

引用本文（GB/T 7714）

DOI：https://doi.org/10.1360/012014-6