Abstract:
In the paper, the general problem of extremal functional interpolation in the mean for real functions that have derivative of order $n$ almost everywhere is formulated on an arbitrary partition $\Delta=\{x_k\}^{\infty}_{k=-\infty}$ of the real axis. It is required to find the smallest value of the $L_{\infty}$-norm of the $n$-derivative among functions that interpolate in the mean (with averaging intervals of length $2h$) any sequence of real numbers $y=\{y_k\}^{\infty}_{k=-\infty}$ from a class $Y$ of sequences whose divided differences of order $n$ are bounded from above on such a grid. In this paper, the problem is considered in the case of $n=2$. We give the above and below estimates for the $L_{\infty}$-norm of the second derivative in terms of grid steps $h_k=x_{k+1}-x_k$ provided that $2h\le \underline{h}=\inf_k h_k$. The obtained results are developments is research of Yu. N. Subbotin, the author and S. I. Novikov in the well-known Yanenko — Stechkin problem of extremal functional interpolation. This problem was put in the early 60-s years of the last century for the case of the uniform grid.
Keywords:interpolation in the mean, derivative, divided difference, axis, grid of nodes, spline.
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