Local Extremal Interpolation on the Semiaxis with the Least Value of the Norm for a Linear Differential Operator

Abstract

On a uniform grid of nodes on the semiaxis $[0;+\infty)$, a generalization is considered of Yu. N. Subbotin’s problem of local extremal functional interpolation of numerical sequences $y=\{y_k\}_{k=0}^\infty$ that have bounded generalized finite differences corresponding to a linear differential operator $\mathscr L_n$ of order $n$ and whose first terms $y_0,y_1,\dots$, $y_{s-1}$ are predefined. Here it is required to find an $n$ times differentiable function $f$ such that $f(kh)=y_k$ $(k\in\mathbb Z_+,h>0)$ which has the least norm of the operator $\mathscr L_n$ in the space $L_\infty$. For linear differential operators with constant coefficients for which all roots of the characteristic polynomial are real and pairwise distinct, it is proved that this least norm is finite only in the case of $s\ge n$.