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 problemof extreme functional interpolation of real-valued functions of one variable (forfinite 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 interpolatingany 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 ofinterpolated sequences for which the sequence of $n$-th order divideddifferences belongs to $l_p(\mathbb Z)$. In the present paper thisproblem is solved in the case when $n=2$. The indicated value is estimated fromabove and below using the greatest and the least step of the grid of nodes.

interpolation, divided difference, spline, difference equation