ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 41 Kryakin, Yuri, Odessa State Polytechnic University, Ukraine Polynomial Interpolation and Whitney’s Constants The well-known theorem of H. Whitney states that For every natural n ≥ 1 and function f ∈ C[a, b] there is an algebraic polynomial P_{n−1} and constant W (n) such that

|f(x) − P_{n−1}(x)| ≤ W (n) sup_{x,x+nh∈[a,b]} |Δ_{h}^{n}f(x)|, x ∈ [a, b].

We obtained a simple proof of Whitney’s theorem and gave estimates W (n) ≤ 2 with the use of polynomial \overline{P}_{n−1} “interpolating on the average”:

∫_{a}^{a+i(b−a)/n} (f(x) − \overline{P}_{n−1}(x)) dx = 0, i = 1, …, n.

Denote by \overline{W} (n) the best constant for approximation by \overline{P}_{n−1}. We conjectured that \overline{W} (n) = 1 and have proved this conjecture for n ≤ 4.