Geodesic
Invariant subspaces of analytic functions. II. Spectral synthesis of convex domains
Sbornik. Mathematics, Tome 17 (1972) no. 1, pp. 1-29 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The criterion for the admissibility of spectral synthesis which was established in the first part of this paper is employed in the solution of a series of problems; in particular, it is employed in the investigation of the homogeneous convolution equation \begin{equation} S*f=0 \tag{\ast} \end{equation} and in the investigation of systems of such equations. Let $H$ be the space of functions holomorphic in a convex region $G$. Let $S$ be a continuous linear functional on $H$. Then the subspace of solutions $f\in H$ of the equation ($\ast$) is invariant and always permits spectral synthesis. However, the system of equations $S_1*f=0,\dots,S_n*f=0$ does not always admit spectral synthesis. In this paper we determine in terms of characteristic functions the precise conditions for the possibility of spectral synthesis for this situation. If $G$ is an unbounded convex region, then spectral synthesis is always possible. Bibliography: 24 titles. 
@article{SM_1972_17_1_a0,
     author = {I. F. Krasichkov-Ternovskii},
     title = {Invariant subspaces of analytic functions. {II.~Spectral} synthesis of convex domains},
     journal = {Sbornik. Mathematics},
     pages = {1--29},
     year = {1972},
     volume = {17},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1972_17_1_a0/}
}
TY  - JOUR
AU  - I. F. Krasichkov-Ternovskii
TI  - Invariant subspaces of analytic functions. II. Spectral synthesis of convex domains
JO  - Sbornik. Mathematics
PY  - 1972
SP  - 1
EP  - 29
VL  - 17
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1972_17_1_a0/
LA  - en
ID  - SM_1972_17_1_a0
ER  - 
%0 Journal Article
%A I. F. Krasichkov-Ternovskii
%T Invariant subspaces of analytic functions. II. Spectral synthesis of convex domains
%J Sbornik. Mathematics
%D 1972
%P 1-29
%V 17
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1972_17_1_a0/
%G en
%F SM_1972_17_1_a0
I. F. Krasichkov-Ternovskii. Invariant subspaces of analytic functions. II. Spectral synthesis of convex domains. Sbornik. Mathematics, Tome 17 (1972) no. 1, pp. 1-29. http://geodesic.mathdoc.fr/item/SM_1972_17_1_a0/

[1] I. F. Krasichkov, “Invariantnye podprostranstva analiticheskikh funktsii. I. Spektralnyi sintez na vypuklykh oblastyakh”, Matem. sb., 87(129) (1972), 459–487

[2] L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz v normirovannykh prostranstvakh, Fizmatgiz, Moskva, 1959 | MR

[3] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, Moskva, 1956

[4] V. Bernstein, Lecons sur les progrès récents de la théorie des séries de Dirichlet, Gauthier-Villars, Paris, 1933 | Zbl

[5] S. Pincherle, “Sur la résolution de l'equation fonctionnelle $\sum h_0\varphi(x+a_0)=f(x)$ a coefficients constants”, Acta Math., 48 (1926), 279–391 | DOI | MR

[6] J. P. Ritt, “On a general class of linear homogeneous differential equations of infinite order with constant coefficients”, Trans. Amer. Math., 18:1 (1917), 27–49 | DOI | MR | Zbl

[7] G. Pólya, “Eine Verallgemeinerung des Fabryschen Lückensatzes”, Nachr. Acad. Wiss. Göttingen, Math.-Phys. Klasse, 2 (1927), 187–195

[8] M. G. Valiron, “Sur les solution des équations differentielles linéaires d'ordre infini et á coefficients constants”, Ann. Ecole Norm. Super., 46 (1929), 25–53 | MR | Zbl

[9] A. O. Gelfond, “Lineinye differentsialnye uravneniya s postoyannymi koeffitsientami i asimptoticheskie periody tselykh funktsii”, Trudy Matem. in-ta im. V. A. Steklova, XXXVIII, 1951, 42–67 | MR

[10] A. F. Leontev, Ryady polinomov Dirikhle i ikh obobscheniya, Trudy Matem. in-ta im. V. A. Steklova, XXXIX, 1951 | MR | Zbl

[11] A. F. Leontev, “O predstavlenii funktsii ryadami polinomov Dirikhle”, Matem. sb., 70(112) (1966), 132–144 | MR | Zbl

[12] D. G. Dickson, “Expansions in series of solutions of linear difference-differential and infinite order differential equations with constant coefficients”, Mem. Amer. Math. Soc., 23, 1967 | MR

[13] D. G. Dickson, “Analytic mean periodic functions”, Trans. Amer. Math. Soc., 110:2 (1964), 361–374 | DOI | MR | Zbl

[14] I. P. Kahane, Sur quelques problémes d'unicité et de prolongement, relatifs aux fonctions approchables par des sommes d'exponentielles, Thèses, Durand, 1954

[15] L. Schwartz, “Théorie générate des fonctions moyenne-périodique”, Ann. Math., 48:4 (1947), 857–929 | DOI | MR | Zbl

[16] H. Viner, R. Peli, Preobrazovanie Fure v kompleksnoi oblasti, Nauka, Moskva, 1964 | MR

[17] M. Cartwright, “On certain integral functions of order 1 and mean type”, Proc. Cambridge Phil. Soc., 31:3 (1935), 347–350 | DOI | MR | Zbl

[18] A. Pfluger, “Uber gewisse ganze Funktionen vom Exponentialtypus”, Comm. Math. Helv., 16:1 (1954–1944), 1–18 | DOI | MR

[19] R. P. Boas, “Asymptotic properties of functions of exponential type”, Duke Math. I, 20:3 (1953), 433–448 | DOI | MR | Zbl

[20] J. P. Kahane, L. A. Rubel, “On Weierstrass products of zero type on the real axis”, Illinois J. Math., 4:4 (1960), 584–592 | MR | Zbl

[21] I. F. Krasichkov, “Otsenki snizu dlya tselykh funktsii konechnogo poryadka”, Sib. matem. zh., 6:4 (1965), 840–861 | MR | Zbl

[22] A. Baillette, “Sur la co-indicatrice des produits canoniques”, Ann. Inst. Fourier, Grenoble, 17:1 (1967), 109–118 | MR

[23] I. F. Krasichkov, “Sravnenie tselykh funktsii konechnogo poryadka po raspredeleniyu ikh kornei”, Matem. sb., 70(112) (1966), 198–230 | Zbl

[24] A. F. Leontev, “O summirovanii ryada Dirikhle s kompleksnymi pokazatelyami i ego primenenii”, Trudy Matem. in-ta im. V. A. Steklova, CXII, 1971, 300–326 | MR