Relative widths of smooth functions determined by fractional order derivatives

For two subsets W and V of a normed space X. The relative Kolmogorov n-width of W relative to V in X is defined by Kn(W, V )X := inf Ln sup f W inf gV Ln f - g X, where the infimum is taken over all n-dimensional subspaces Ln of X. For R+, define Wp (1 p ) to be the collection of 2 -periodic and continuous functions f representable as a convolution f (t) = c + (B g)(t), where g Lp(T ), T = [0, 2 ], g p 1, T g(x)dx = 0, and B (t) L1(T ) with the Fourier expanded form B (t) = 1 2 kZ\{0} (ik)- eikt . In this article, we discuss the relative Kolmogorov n-width of Wp relative to Wp in the space Lq(T ). For the case p=, 1 q , and the case p=1, 1 q 2, > 1- 1 q and the case p=1, 2 < q , > 3- 1 q , we obtain their weak asymptotic results. In addition, we also obtain the weak asymptotic result of Wp relative to Wp in the space Lp(T ) for 0 < 2.

• Type: Journal
• Year: 2008
• Where: JC
• Authors: Liu Yongping, Yang Lianhong
• Keywords: JC 2008, Ln Sup, Normed Space X, Relative Kolmogorov