Geodesic
Spectral synthesis and analytic continuation
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 1, pp. 31-108 Cet article a éte moissonné depuis la source Math-Net.Ru

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A closed subspace of functions holomorphic in a domain of the $n$-dimensional complex space is considered. It is assumed that the subspace is invariant with respect to the partial differentiation operators and admits spectral synthesis, that is, coincides with the closure of the linear span of the common root elements in it of the partial differentiation operators. Conditions under which the elements of the invariant subspace admit analytic continuation to a larger domain are studied. The geometry of this domain depends both on the original domain and on the existence of functions admitting special lower bounds in the annihilator submodule of the invariant subspace. The same problem is also considered for topological products of invariant subspaces. The results are applied to the analytic continuation of solutions of homogeneous convolution equations. 
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I. F. Krasichkov-Ternovskii. Spectral synthesis and analytic continuation. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 1, pp. 31-108. http://geodesic.mathdoc.fr/item/RM_2003_58_1_a1/

[1] L. Schwartz, “Théorie générale des fonctions moyenne-périodique”, Ann. of Math. (2), 48:4 (1947), 857–929 | DOI | MR | Zbl

[2] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. I: Spektralnyi analiz na vypuklykh oblastyakh”, Matem. sb., 87(129):4 (1972), 459–489 | MR

[3] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. II: Spektralnyi sintez na vypuklykh oblastyakh”, Matem. sb., 88(130):1 (1972), 3–30

[4] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. III: O rasprostranenii spektralnogo sinteza”, Matem. sb., 88(130):3 (1972), 331–352 | MR

[5] I. F. Krasichkov-Ternovskii, “Spektralnyi sintez analiticheskikh funktsii na sistemakh vypuklykh oblastei”, Matem. sb., 111(153):1 (1980), 3–41 | MR

[6] I. F. Krasichkov-Ternovskii, “Spektralnyi sintez analiticheskikh funktsii na sistemakh vypuklykh oblastei. Rasprostranenie sinteza”, Matem. sb., 112(154):1 (1980), 94–114 | MR

[7] I. F. Krasichkov-Ternovskii, “Spektralnyi sintez analiticheskikh funktsii na sistemakh neogranichennykh vypuklykh oblastei”, Matem. sb., 111(153):3 (1980), 384–401 | MR

[8] I. F. Krasichkov-Ternovskii, “Dlya kazhdogo invariantnogo podprostranstva, dopuskayuschego spektralnyi sintez, suschestvuet metod approksimatsii”, Sib. matem. zhurn., 22(154):3 (1981), 74–90 | MR

[9] I. F. Krasičkov-Ternovskii, “Spectral synthesis on a system of unbounded domains starlike in a common direction”, Anal. Math., 19:3 (1993), 217–223 | DOI | MR | Zbl

[10] B. Malgrange, “Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution”, Ann. Inst. Fourier (Grenoble), 6 (1955–1956), 271–355 | MR

[11] L. Ehrenpreis, “Mean periodic functions. I: Varieties whose annihilator ideals are principal”, Amer. J. Math., 77 (1955), 293–328 | DOI | MR | Zbl

[12] R. S. Yulmukhametov, “Odnorodnye uravneniya svertki”, Dokl. AN SSSR, 316:2 (1991), 312–315 | MR | Zbl

[13] A. S. Krivosheev, V. V. Napalkov, “Kompleksnyi analiz i operatory svertki”, UMN, 47:6 (1992), 3–58 | MR | Zbl

[14] A. B. Shishkin, “Spektralnyi sintez dlya sistem differentsialnykh operatorov s postoyannymi koeffitsientami. Teorema dvoistvennosti”, Matem. sb., 189:9 (1998), 143–160 | MR | Zbl

[15] I. F. Krasichkov-Ternovskii, “Spektralnyi sintez i lokalnoe opisanie dlya mnogikh peremennykh”, Izv. RAN. Ser. matem., 63:4 (1999), 101–130 | MR

[16] V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, Nauka, M., 1982 | MR

[17] A. S. Krivosheev, “Analiticheskoe prodolzhenie funktsii iz invariantnykh podprostranstv v vypuklykh oblastyakh kompleksnogo prostranstva”, Izv. RAN. Ser. matem., 62:2 (1998), 77–102 | MR

[18] L. Ehrenpreis, “Appendix to the paper: “Mean periodic functions I””, Amer. J. Math., 77 (1955), 731–733 | DOI | MR | Zbl

[19] J. Hadamard, “Essai sur l'étude des fonctions donnés par leur développement de Taylor”, J. Math. Pures Appl. Sér. (4), 4 (8) (1892), 101–106

[20] E. Fabry, “Sur les points singuliers d'une fonction donnée par son développement de Taylor”, Ann. École Norm. Sup. (3), 13 (1896), 367–399 | MR | Zbl

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

[22] A. Ostrowski, “Über die analytishe Fortsetzung von Taylorshen und Dirichletshen Reihen”, Math. Ann., 129 (1955), 1–43 | DOI | MR | Zbl

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

[24] G. Pólya, “Über die Existenz unendlich vieler singulärer Punkte auf der Konvergenzgeraden gewisser Dirichlet'sher Reihen”, Sitzungber. Preuß. Akad. Wiss., 1923, 45–50 | Zbl

[25] A. F. Leontev, “O klasse funktsii, opredelennykh ryadom polinomov Dirikhle”, UMN, 3:4 (1948), 3–58 | MR

[26] A. F. Leontev, Ryady polinomov Dirikhle i ikh obobscheniya, Trudy MIAN, 39, 1951 | MR | Zbl

[27] J.-P. Kahane, “Sur quelques problèmes d'unicité et de prolongement, relatifs aux fonctions approchables par des sommes d'exponentielles”, Ann. Inst. Fourier (Grenoble), 5 (1953–1954), 39–130 | MR

[28] A. F. Leontev, “O skhodimosti posledovatelnostei polinomov Dirikhle”, Dokl. AN SSSR, 108:1, 23–26 | Zbl

[29] A. F. Leontev, “Novoe dokazatelstvo odnoi teoremy o skhodimosti posledovatelnostei polinomov Dirikhle”, UMN, 12:3 (1957), 165–170 | MR | Zbl

[30] A. F. Leontev, “O svoistvakh posledovatelnostei polinomov Dirikhle, skhodyaschikhsya na intervale mnimoi osi”, Izv. AN SSSR. Ser. matem., 29:2 (1965), 269–328 | MR | Zbl

[31] L. Schwartz, Étude des sommes d'exponentielles, Hermann, Paris, 1959 | MR

[32] A. Baillette, “Approximation de fonctions par des sommes d'exponentielles”, C. R. Acad. Sci. Paris, 249:23 (1959), 2470–2471 | MR | Zbl

[33] A. Bailette, “Fonctions approchables par des sommes d'exponentielles”, J. Anal. Math., 10 (1962–1963), 91–115 | DOI | MR

[34] I. F. Krasichkov, “O skhodimosti polinomov Dirikhle”, Sib. matem. zhurn., 7:5 (1966), 1039–1058 | MR | Zbl

[35] I. F. Krasichkov-Ternovskii, “Invariantnye podprostranstva analiticheskikh funktsii. Analiticheskoe prodolzhenie”, Izv. AN SSSR. Ser. matem., 37:4 (1973), 931–945 | MR

[36] L. Khërmander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[37] P. Schapira, “Équations aux dérivées partielles dans l'espace des hyperfonctions”, Lecture Notes in Math., 71, 1968, 38–45 | MR | Zbl

[38] C. O. Kiselman, “Prolongement des solutions d'une équation aux dérivées partielles à coefficients constants”, Bull. Soc. Mat. France, 97 (1969), 329–356 | MR | Zbl

[39] L. Ehrenpreis, Fourier analysis in several complex variables, Pure Appl. Math., 17, Wiley, New York, 1970 | MR | Zbl

[40] V. P. Palamodov, Lineinye differentsialnye operatory s postoyannymi koeffitsientami, Nauka, M., 1967 | MR

[41] A. Sebbar, “Prolongement des solutions holomorphes de certains opérateurs différentiels d'ordre infini à coefficients constants”, Lecture Notes in Math., 822, 1980, 199–220 | MR | Zbl

[42] A. Meril, D. S. Struppa, “Convolutors in spaces of holomorphic functions”, Lecture Notes in Math., 1276 (1987), 253–275 | DOI | MR | Zbl

[43] A. S. Krivosheev, “Ob indikatore tselykh funktsii i prodolzhenii reshenii odnorodnogo uravneniya svertki”, Matem. sb., 184:8 (1993), 81–108 | Zbl

[44] Zh. Sebastyan-i-Silva, “O nekotorykh klassakh lokalno-vypuklykh prostranstv, vazhnykh v prilozheniyakh”, Matematika. Sb. perevodov, 1:1 (1957), 60–77 | MR

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

[46] I. F. Krasichkov-Ternovskii, “Lokalnoe opisanie zamknutykh idealov i podmodulei analiticheskikh funktsii odnoi peremennoi. I”, Izv. AN SSSR. Ser. matem., 43:1 (1979), 44–66 | MR

[47] H. Cartan, “Idéaux et modules de fonctions analytiques de variables complexes”, Bull. Soc. Math. France, 78:4 (1950), 29–64 | MR | Zbl

[48] Zh. Dedonne, L. Shvarts, “Dvoistvennost v prostranstvakh $(F)$ i $(LF)$”, Matematika. Sb. perevodov, 2:2 (1958), 77–102

[49] R. Rokafellar, Vypuklyi analiz, Mir, M., 1973

[50] V. G. Boltyanskii, “O nekotorykh klassakh vypuklykh mnozhestv”, Dokl. AN SSSR, 226:1 (1976), 19–22 | MR | Zbl

[51] V. G. Boltyanskii, “Teoriya Khelli dlya $H$-vypuklykh mnozhestv”, Dokl. AN SSSR, 226:2 (1976), 249–252 | MR | Zbl

[52] L. Dantser, B. Gryunbaum, V. Kli, Teorema Khelli, Mir, M., 1968

[53] V. G. Boltyanskii, P. S. Soltan, Kombinatornaya geometriya razlichnykh klassov vypuklykh mnozhestv, Shtiintsa, Kishinev, 1978 | MR | Zbl

[54] V. G. Boltyanskii, E. D. Baladze, Problema Sekefalvi-Nadya v kombinatornoi geometrii, Nauka, M., 1997 | MR | Zbl

[55] R. Engelking, Obschaya topologiya, Mir, M., 1986 | MR

[56] L. S. Pontryagin, “Lineinye differentsialnye igry presledovaniya”, Matem. sb., 112:3 (1980), 307–330 | MR | Zbl

[57] S. Banach, Théorie des opérations linéaires, Subwncji Funduszu Narodowej, Warszawa, 1932 | Zbl

[58] A. Robertson, V. Robertson, Topologicheskie vektornye prostranstva, Mir, M., 1967 | MR | Zbl

[59] N. S. Landkof, Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl