@MISC{Shadrin_thelandau–kolmogorov, author = {Alexei Shadrin}, title = {THE LANDAU–KOLMOGOROV INEQUALITY REVISITED}, year = {} }

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Abstract

Abstract. We consider the Landau–Kolmogorov problem on a finite interval which is to find an exact bound for ‖f (k) ‖, for 0 < k < n, given bounds ‖f ‖ ≤ 1 and ‖f (n) ‖ ≤ σ, with ‖ · ‖ being the max-norm on [−1, 1]. In 1972, Karlin conjectured that this bound is attained at the end-point of the interval by a certain Zolotarev polynomial or spline, and that was proved for a number of particular values of n or σ. Here, we provide a complete proof of this conjecture in the polynomial case, i.e. for 0 ≤ σ ≤ σn: = ‖T (n) n ‖, where Tn is the Chebyshev polynomial of degree n. In addition, we prove a certain Schur-type estimate which is of independent interest. 1. Introduction. The Landau–Kolmogorov (LK-