Abstract:

For two subsets W and V of a Banach space X, let K_n(W,V,X) denote the relative Kolmogorov n-width of W relative to V defined by

K_n(W,V,X):= inf_Ln sup_f∈W inf _{g∈ V∩ Ln} || f-g||_X,
where the infimum is taken over all n-dimensional linear subspaces L_n of X. Let W₂(△^r) denote the class of 2π-periodic functions f with d-variables satisfying
∫_[-π,π]^d |△^r f(x)|²,dx≤ 1,
while △^r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W₂(△^r)$ relative to W₂(△^r) in L_q([-π,π]^d),(1≤ q≤∞), and obtain its weak asymptotic result.

Key words: Multivariate function classes, width, relative width