Sample numbers and optimal Lagrange interpolation in Sobolev spaces

Abstract

This paper investigates the optimal recovery of Sobolev spaces W∞r[−1,1],r∈ℕ in space L∞[−1,1] and weighted spaces Lp,ω[−1,1],1≤p<∞ with ω a continuous integrable weight function in (−1,1). We obtain the values of the sampling numbers of W∞r[−1,1] in L∞[−1,1] and Lp,ω[−1,1],1≤p<∞. We prove that the Lagrange interpolation algorithms based on the Chebyshev nodes of the first kind are optimal for p=∞. Meanwhile, we prove that the Lagrange interpolation algorithms based on the zeros of polynomial of degree r with the leading coefficient 1 of the least deviation from zero in Lp,ω[−1,1] are optimal for 1≤p<∞. We also give the optimal Lagrange interpolation algorithms when we ask the endpoints to be included in the nodes.