最坏框架与平均框架下区间[-1, 1]上带Jacobi权的函数逼近

Abstract

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 d n ( BW p,α,β r , L q,α,β ) and linear n -widths δ n ( BW p,α,β r , L q,α,β ) of the weighted Sobolev classes BW p,α,β on [-1,1], where L q,α,β ,1 ≤ q ≤∞ denotes the L q space on [-1,1] with respect to Jocobi weights. Optimal asymptotic orders of d n ( BW p,α,β r , L q,α,β ) and δ n ( BW p,α,β r , L q,α,β ) as n →∞ are obtained for all 1 ≤ p,q ≤ ∞. In the average case setting, we investigate the best approximation of functions on the weighted Sobolev class B W 2, α,β r equipped with a centered Gaussian measure by polynomial subspaces in the L q,α,β metric for 1 ≤ q L q,α,β metric only for 1 q α,β }+ 1).