RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC*

For two subsets W and V of a Banach space X, let Kn (W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn(W,V,X)=infLnsupf∈ Winfg∈V∩Ln∥f-g∥x, where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(Δr) denote the class of 2π-periodic functions f with d-variables satisfying ∫d[-π,π]|Δrf(x)|2dx≤1,∫[-π,π]d|Δrf(x)|2dx≤1, while Δr is the r-iterate of Laplace operator Δ. This article discusses the relative Kolmogorov n-width of W2(Δr) relative to W2(Δr) in Lq ([−π, π]d) (1 ≤ q ≤ ∞), and obtain its weak asymptotic result.