Relative widths of smooth functions determined by fractional order derivatives

For two subsetsW and V of a normed space X. The relative Kolmogorov n-width ofW relative to V in X is defined by Kn(W,V )X := inf Ln sup f∈W inf g∈V∩Ln ‖f − g‖X , where the infimum is taken over all n-dimensional subspaces Ln of X. For ∈ R+, defineW p (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 ∑ k∈Z\{0} (ik)− e . In this article, we discuss the relative Kolmogorov n-width ofW p relative toW p in the spaceLq(T ). For the casep=∞, 1 q ∞, and the casep=1, 1 q 2, > 1− 1 q and the casep=1, 2 3− 1 q , we obtain their weak asymptotic results. In addition, we also obtain the weak asymptotic result of W p relative toW p in the space Lp(T ) for 0< 2. © 2007 Elsevier Inc. All rights reserved.