Geodesic
On zero sets of weakly localisable pricipal submodules in the Schwartz algebra
Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 3, pp. 261-270.

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

We consider the Schwartz algebra $\mathcal P.$ As a linear topological space, it is isomorphic to the space of all distributions compactly supported on the real line. By the Paley — Wiener — Schwartz theorem, the Fourier — Laplace transform establishes the corresponding isomorphism. Submodules of the algebra $\mathcal P$ are defined as closed subspaces which are invariant under the multiplication by the independent variable $z.$ They supply an effective tool to explore the possibility of the spectral synthesis for the differentiation operator in the space $C^{\infty} (\mathbb R).$ In connection with some open questions on the problem of the spectral synthesis in $C^{\infty} (\mathbb R)$, we study principal submodules of the algebra $\mathcal P.$ Earlier, we have obtained the sufficient conditions and the weighted criterion of the weak localisability for principal submodules. These conditions contain some restrictions on the generating function of a submodule. However, one should also consider the following form of the question: knowing the zero set of a principal submodule (or, which is the same, the zero set of its generating function), define whether it is weakly localisable. The complete answer seems to be quite difficult to find. Here, we construct the class of synthesable sequences which are zero sets of weakly localisable principal submodules.
Keywords: entire function, zero set, Schwartz algebra, spectral synthesis, localisable submodule. 
@article{CHFMJ_2020_5_3_a0,
     author = {N. F. Abuzyarova and A. F. Sagadieva and Z. Yu. Fazullin},
     title = {On zero sets of weakly localisable pricipal submodules in the {Schwartz} algebra},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {261--270},
     publisher = {mathdoc},
     volume = {5},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_3_a0/}
}
TY  - JOUR
AU  - N. F. Abuzyarova
AU  - A. F. Sagadieva
AU  - Z. Yu. Fazullin
TI  - On zero sets of weakly localisable pricipal submodules in the Schwartz algebra
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2020
SP  - 261
EP  - 270
VL  - 5
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_3_a0/
LA  - ru
ID  - CHFMJ_2020_5_3_a0
ER  - 
%0 Journal Article
%A N. F. Abuzyarova
%A A. F. Sagadieva
%A Z. Yu. Fazullin
%T On zero sets of weakly localisable pricipal submodules in the Schwartz algebra
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2020
%P 261-270
%V 5
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_3_a0/
%G ru
%F CHFMJ_2020_5_3_a0
N. F. Abuzyarova; A. F. Sagadieva; Z. Yu. Fazullin. On zero sets of weakly localisable pricipal submodules in the Schwartz algebra. Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 3, pp. 261-270. http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_3_a0/

[1] Sebastiao et Silva J., “Su certe classi di spazi localmente convessi importanti per le applicazioni”, Rendiconti di Matematica e delle sue Applicazioni. Roma, 14:5 (1955), 388–410 | MR | Zbl

[2] B. Ya. Levin, Lectures on entire functions, Rev. Edition, AMS Providence, R. I., 1996 (with Yu. Lubarskii, M. Sodin, V. Tkachenko), 254 pp. | MR | Zbl

[3] Abuzyarova N.F., “Spectral synthesis in a Schwartz space of infinitely differentiable functions”, Doklady Mathematics, 90:1 (2014), 479–482 | DOI | MR | Zbl

[4] A. Aleman, B. Korenblum, “Derivation-invariant subspaces of $C^{\infty}$”, Computational Methods and Function Theory, 8:2 (2008), 493–512 | DOI | MR | Zbl

[5] A. Aleman, A. Baranov, Yu. Belov, “Subspaces of $C^{\infty}$ invariant under the differentiation”, Journal of Functional Analysis, 268 (2015), 2421–2439 | DOI | MR | Zbl

[6] Abuzyarova N.F., “Spectral synthesis for the differentiation operator in the Schwartz space”, Mathematical Notes, 102:2 (2017), 137–148 | DOI | MR | MR | Zbl

[7] A. Baranov, Yu. Belov, “Synthesizable differentiation-invariant subspaces”, Geometric and Functional Analysis, 29:1 (2019), 44–71 | DOI | MR | Zbl

[8] Abuzyarova N.F., “Closed submodules in the module of entire functions of exponential type and polynomial growth on the real axis”, Ufa Mathematical Journal, 6:4 (2014), 3–17 | DOI | MR

[9] Abuzyarova N.F., “Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis”, Ufa Mathematical Journal, 8:1 (2016), 1–12 | DOI | MR

[10] L. Ehrenpreis, “Solution of some problems of division. IV”, American Journal of Mathematics, 57 (1960), 522–588 | DOI | MR

[11] N. F. Abuzyarova, “Principal submodules in the module of entire functions, which is dual to the Schwarz space, and weak spectral synthesis in the Schwartz space”, Journal of Mathematical Sciences, 241:6 (2019), 658–671 | DOI | MR | Zbl

[12] Abuzyarova N.F., “Principal submodules in the Schwartz module”, Russian Mathematics, 64 (2020), 74–78 | DOI

[13] Abuzyarova N.F., “On 2-generateness of weakly localizable submodules in the module of entire functions of exponential type and polynomial growth on the real axis”, Ufa Mathematical Journal, 8:3 (2016), 8–21 | DOI | MR

[14] Abuzyarova N.F., “On shifts of a sequence of integers generating functions that are invertible in the sense of Ehrenpreis”, Scientific seminars notes of PDMI, 480 (2019), 5–25 (In Russ.)

[15] Krasnoselsky M.A., Rutitsky Y.B., Convex Functions and Orlicz Spaces, Hindustan Publ., 1962, 262 pp. | MR

[16] R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954, 276 pp. | MR | Zbl