Geodesic
Nonlocal almost differential operators and interpolation by functions with sparse spectrum
Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 131-140 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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/

[1] De Branges L., “Local operators on Fourier transforms”, Zap. nauchn. sem. LOMI, 81 (1978), 248

[2] Arutyunyan F. G., “Predstavlenie funktsii kratnymi ryadami”, DAN ArmSSR, 64:2 (1977), 72–76 | MR | Zbl

[3] Menshov D. E., “O ravnomernoi skhodimosti ryadov Fure”, Matem. sb., 11(53) (1942), 67–96 | Zbl

[4] Kisliakov S. V., “Un théoréme de correction de type Menšov”, C. R. Acad. Sci. Paris, 296 (1983), 39–42 | MR | Zbl

[5] De Branges L., Hilbert spaces of entire functions, New Jersey, 1968

[6] Beurling A., Lectures on quasi-analiticity, A. M. S. Summer Institute, Stanford, 1967

[7] Beurling A., Malliavin P., “On Fourier transforms of measures with compact support”, Acta Math., 1962, no. 107, 291–309 | DOI | MR | Zbl