Geodesic
Extremal Interpolation in the Mean in the Space~$L_1(\mathbb R)$ with Overlapping Averaging Intervals
Matematičeskie zametki, Tome 115 (2024) no. 1, pp. 123-136

Voir la notice de l'article provenant de la source Math-Net.Ru

On a uniform grid on the real axis $\mathbb R$, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space $L_1(\mathbb R)$ of two-way real sequences with the least value of the norm of a linear formally self-adjoint differential operator $\mathcal L_n$ of order $n$ with constant real coefficients. This problem is considered for the class of sequences for which the generalized finite differences of order $n$ corresponding to the operator $\mathcal L_n$ are bounded in the space $l_1$. In this paper, the least value of the norm is calculated exactly if the grid step $h$ and the averaging step $h_1$ of the function to be interpolated in the mean are related by the inequalities $h$. The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the $n$-times differentiation operator $\mathcal L_n(D)=D^n$.
Keywords: extremal interpolation in the mean, spline, uniform grid, formally self-adjoint differential operator, least norm. 
@article{MZM_2024_115_1_a8,
     author = {V. T. Shevaldin},
     title = {Extremal {Interpolation} in the {Mean} in the {Space~}$L_1(\mathbb R)$ with {Overlapping} {Averaging} {Intervals}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {123--136},
     publisher = {mathdoc},
     volume = {115},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2024_115_1_a8/}
}
TY  - JOUR
AU  - V. T. Shevaldin
TI  - Extremal Interpolation in the Mean in the Space~$L_1(\mathbb R)$ with Overlapping Averaging Intervals
JO  - Matematičeskie zametki
PY  - 2024
SP  - 123
EP  - 136
VL  - 115
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2024_115_1_a8/
LA  - ru
ID  - MZM_2024_115_1_a8
ER  - 
%0 Journal Article
%A V. T. Shevaldin
%T Extremal Interpolation in the Mean in the Space~$L_1(\mathbb R)$ with Overlapping Averaging Intervals
%J Matematičeskie zametki
%D 2024
%P 123-136
%V 115
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2024_115_1_a8/
%G ru
%F MZM_2024_115_1_a8
V. T. Shevaldin. Extremal Interpolation in the Mean in the Space~$L_1(\mathbb R)$ with Overlapping Averaging Intervals. Matematičeskie zametki, Tome 115 (2024) no. 1, pp. 123-136. http://geodesic.mathdoc.fr/item/MZM_2024_115_1_a8/