Universal interpolation

摘要

If P_n is the polynomial of degree at most n - 1 which interpolates a function f : [0,1] → R at the nodes 0 ≤ x_1~n ≤ x_2~n < ··· < x_n~n ≤ 1 (n ∈ N), it is well-known that, even if f is a continuous function, the sequence (P_n)n ∈ N does not necessarily converge to /. Indeed, for p 6 [1, ∞), there exists an infinitely often differentiable function / and a "nice" system of nodes such that to every measurable function g, there exists a subsequence of (P_n)n∈N that converges in L_p to g.