Sharp Estimates of High-Order Derivatives in Sobolev Spaces

The paper describes the splines Q(n,k)(x, a), which define the relations y((k))(a) = integral(1)(0) y((n))(x)Q(n,k)((n)) (x, a)dx for an arbitrary point alpha is an element of(0; 1) and an arbitrary function y is an element of W-p(n)[0; 1]. The connection of the minimization of the norm ||Q(n,k)((n))|| L-p' [0;1] (1/p + 1/ p' = 1) by parameter a with the problem of best estimates for derivatives |y((k))(a)| <= A(n, k,p)(a)||y((n)) || L-p[0;1], and also with the problem of finding the exact embedding constants of the Sobolev space W-p(n)[0; 1] into the space W-infinity(k) [0; 1], n is an element of N, 0 <= k <= n - 1. Exact embedding constants are found for all n is an element of N, k = n - 1 for p = 1 and for p = infinity.