Kolmogorov-Landau Functions

‖f ‖ ≤ Crq ‖f‖ 1−q/r ‖f ‖ was proved by Kolmogorov in [11] for f ∈W r ∞(R). Here ‖ · ‖ means ‖ · ‖L∞(R) and the sharp constant Crq = Kr−qK −1+q/r r is given in terms of the Favard constants Kp := 4 π ∑∞ n=0 [ (−1) 2n+1 p+1 . For the definition of the Sobolev space W r ∞(R) see e.g. [4, Ch. 1.5]. The first results of this type (the case q = 1, r = 2) were obtained by Hadamard in [9] and by Landau in [12] (for ‖ · ‖L∞(R+)), who initiated this class of extremal problems. Shilov [14] found the exact constants in the case r ≤ 4, q ≤ r and predicted the class of extremal functions (Euler’s splines) for the inequality in the general case. His conjecture was confirmed by A.N. Kolmogorov. After that, inequalities of Kolmogorov type, especially versions for a finite interval, attracted attention of many authors. For a contribution by Bojanov, see [1, 2, 3]. Let us consider the additive form of the Kolmogorov inequality