For functions from the Sobolev space Wⁿ₀, 1 and an arbitrary point a (0, 1), we study sharp estimating functions A₍, ₊, (a) in the inequality |f^ (k) (a) | A₍, ₊, (a) \|f^ (n) \|₋_ ₀, ₁, 0 k < n. We establish a link between the functions A₍, ₊, and best approximations of special splines by polynomials in L₁0, 1. We introduce the Markov set A₍, ₊, 0 k < n, of values of the parameter a, on which we obtain the representation A₍, ₊, (a) = 2^- (n-k) |Vₙ^ (k) (2a-1) | of the function A₍, ₊, in terms of the absolute value of the k th derivative of the Peano kernel Vₙ of order n. For arbitrary n and k, 0 k n-2, we show that the embedding constant ₍, ₊, of the Sobolev spaces Wⁿ₀, 1 Wᵏ₀, 1 is the maximum value of the function 2^- (n-k) |Vₙ^ (k) (2a-1) | on the interval 0, 1. For odd n and even k, 0 k < n, the embedding constant ₍, ₊, is refined as ₍, ₊, =2^- (n-k) |Vₙ^ (k) (0) |. We also express the embedding constants ₍, ₊, for odd n and even k in terms of hypergeometric functions and study the asymptotic behavior of the embedding constants as n=2m+1 with fixed k or n-k.
Kazimirov et al. (Mon,) studied this question.