Let $k$ be a measurable function on $\mathbf R$. Define an operator $\mathscr L_k\colon f\to\mathscr F^{-1}(k\mathscr F(f))$, where $f\in L^2(\mathbf R)$ and $\mathscr F$ is the Fourier transform. Let $\mathscr D_k=\{f\in L^2(\mathbf R):k\mathscr F(f)\in L^2(\mathbf R)\}$ be its domain. The operator $\mathscr L_k$ is called local if $f|E=0$ implies $\mathscr L_k(f)|E=0$ for $E\subset\mathbf R$ with $\operatorname{mes} E>0$. An entire function $g$ of order zero is constructed for which the operator $\mathscr L_g$ is not local. Let $W$ be the Wiener algebra of absolutely convergent trigonometric series. We prove a theorem on correction in the spirit of Luzin's theorem: a condition is exhibited on a set $A$ of integers under which each function of $W$ can be corrected on a set of arbitrarily small measure so that the spectrum of the corrected function (also in $W$) is contained in $A$. Bibliography: 7 titles.
@article{SM_1987_56_1_a8,
author = {P. P. Kargaev},
title = {Nonlocal almost differential operators and interpolation by functions with sparse spectrum},
journal = {Sbornik. Mathematics},
pages = {131--140},
year = {1987},
volume = {56},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1987_56_1_a8/}
}
TY - JOUR
AU - P. P. Kargaev
TI - Nonlocal almost differential operators and interpolation by functions with sparse spectrum
JO - Sbornik. Mathematics
PY - 1987
SP - 131
EP - 140
VL - 56
IS - 1
UR - http://geodesic.mathdoc.fr/item/SM_1987_56_1_a8/
LA - en
ID - SM_1987_56_1_a8
ER -
%0 Journal Article
%A P. P. Kargaev
%T Nonlocal almost differential operators and interpolation by functions with sparse spectrum
%J Sbornik. Mathematics
%D 1987
%P 131-140
%V 56
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1987_56_1_a8/
%G en
%F SM_1987_56_1_a8
P. P. Kargaev. Nonlocal almost differential operators and interpolation by functions with sparse spectrum. Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 131-140. http://geodesic.mathdoc.fr/item/SM_1987_56_1_a8/
