Abstract
For functions from the Sobolev space \(\mathring{W}^n_\infty[0,1]\) and an arbitrary point \(a\in(0,1)\) , we study sharp estimating functions \(A_{n,k,\infty}(a) \) in the inequality \(|f^{(k)}(a)|\le A_{n,k,\infty}(a) \cdot \|f^{(n)}\|_{L_\infty [0,1]}\) , \(0 \le k < n\) . We establish a link between the functions \(A_{n,k,\infty}\) and best approximations of special splines by polynomials in \(L_1[0,1]\) . We introduce the Markov set \(\mathcal{A}_{n,k}\) , \(0 \le k < n\) , of values of the parameter \(a\) , on which we obtain the representation \(A_{n, k, \infty}(a)= 2^{-(n-k)}|V_n^{(k)}(2a-1)|\) of the function \(A_{n,k,\infty}\) in terms of the absolute value of the \(k\) th derivative of the Peano kernel \(V_n\) of order \(n\) . For arbitrary \(n\) and \(k\) , \(0 \le k \le n-2\) , we show that the embedding constant \(\Lambda_{n,k,\infty}\) of the Sobolev spaces \(\mathring{W}^n_\infty[0,1] \hookrightarrow \mathring{W}^k_\infty[0,1]\) is the maximum value of the function \(2^{-(n-k)}|V_n^{(k)}(2a-1)| \) on the interval \([0,1]\) . For odd \(n\) and even \(k\) , \(0 \le k < n\) , the embedding constant \(\Lambda_{n,k,\infty}\) is refined as \(\Lambda_{n,k,\infty}=2^{-(n-k)}|V_n^{(k)}(0)|\) . We also express the embedding constants \(\Lambda_{n,k,\infty}\) for odd \(n\) and even \(k\) in terms of hypergeometric functions and study the asymptotic behavior of the embedding constants as \(n=2m+1 \to \infty\) with fixed \(k\) or \(n-k\) .
