Abstract
We pose the general problem of extremal interpolation in the mean of real functions whose \(n\) th derivative belongs to the space \(L_p(\mathbb R)\) , \(1\le p\le \infty\) . The problem is to find the smallest value of this derivative in \(L_p(\mathbb R)\) for functions \(f\) which interpolate in the mean (with averaging length \(2h\) ) any given sequence of real numbers from the class of sequences introduced by Golomb and de Bohr defined by divided differences of order \(n\) and spacings \(h_k=x_{k+1}-x_k\) of a grid \(\Delta\) . In the present paper, this problem is solved for \(n=2\) , \(p=1\) , and \(0<h<\underline{h}=\inf_k h_k\) .
