Extremal interpolation with the least value of the norm of the second derivative in $L_p(\mathbb R)$

In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the $n$-th derivative in $L_p(\mathbb R)$, $1\le p\le \infty$, needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis $\mathbb R$ for the class of interpolated sequences for which the sequence of $n$-th order divided differences belongs to $l_p(\mathbb Z)$. In the present paper this problem is solved in the case when $n=2$. The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.