Optimal Birkhoff Interpolation and Birkhoff Numbers in Some Function Spaces

This paper investigates the optimal Birkhoff interpolation and Birkhoff numbers of some function spaces in space L ∞ [−1, 1] and weighted spaces L p,ω [−1,1], 1 ≤ p < ∞, with ω being a continuous integrable weight function in (−1,1). We proved that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal. We also show that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal when the function values of the two endpoints are included in the interpolation systems.