Criterion of analytic continuability of functions in principal invariant subspaces on convex domains in ℂⁿ

Krivosheev, A.

doi:10.1090/S1061-0022-2011-01160-4

Criterion of analytic continuability of functions in principal invariant subspaces on convex domains in $\mathbb {C}^{n}$
HTML articles powered by AMS MathViewer

by A. S. Krivosheev
Translated by: the author

St. Petersburg Math. J. 22 (2011), 615-655

DOI: https://doi.org/10.1090/S1061-0022-2011-01160-4

Published electronically: May 3, 2011

PDF | Request permission

Abstract:

Subspaces invariant under differentiation are studied for spaces of functions analytic on domains of a many-dimensional complex space. For a wide class of domains (in particular, for arbitrary bounded convex domains), a criterion of analytic continuability is obtained for functions in arbitrary nontrivial closed principal invariant subspaces admitting spectral synthesis.

References

V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, “Nauka”, Moscow, 1982 (Russian). MR 678923
J. Hadamard, Essai sur l’étude des fonctions données par leur développement de Taylor, J. Math. Pures Appl. Ser. 4 8 (1892), 101–106.
Eugène Fabry, Sur les points singuliers d’une fonction donnée par son développement en série et l’impossibilité du prolongement analytique dans des cas très généraux, Ann. Sci. École Norm. Sup. (3) 13 (1896), 367–399 (French). MR 1508933
I. F. Krasičkov-Ternovskiĭ, Invariant subspaces of analytic functions. I, II. Spectral synthesis on convex domains, Mat. Sb. (N.S.) 87(129) (1972), 459–489; ibid. (N.S.) 88(130) (1972), 3–30 (Russian). MR 0422636
I. F. Krasičkov-Ternovskiĭ, Invariant subspaces of analytic functions. I, II. Spectral synthesis on convex domains, Mat. Sb. (N.S.) 87(129) (1972), 459–489; ibid. (N.S.) 88(130) (1972), 3–30 (Russian). MR 0422636
R. S. Yulmukhametov, Homogeneous convolution equations, Dokl. Akad. Nauk SSSR 316 (1991), no. 2, 312–315 (Russian); English transl., Soviet Math. Dokl. 43 (1991), no. 1, 101–103. MR 1100598
A. S. Krivosheev and V. V. Napalkov, Complex analysis and convolution operators, Uspekhi Mat. Nauk 47 (1992), no. 6(288), 3–58 (Russian); English transl., Russian Math. Surveys 47 (1992), no. 6, 1–56. MR 1209144, DOI 10.1070/RM1992v047n06ABEH000954
I. F. Krasichkov-Ternovskiĭ, Spectral synthesis and analytic continuation, Uspekhi Mat. Nauk 58 (2003), no. 1(349), 33–112 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 1, 31–108. MR 1992131, DOI 10.1070/RM2003v058n01ABEH000593
G. Polya, Uber die Exstenz unendlich vieler singularer Punkte auf der Konvergenzgeraden gewisser Dirichlet’sher Reihen, Sitzungber. Preu. Akad. Wiss. 1923, 45–50.
—, Eine Verallgemeinerung des Fabryschen Lückensatzes, Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl. 2 (1927), 187–195.
V. Bernstein, Leçons sur les progress de la théorie des series de Dirichlet, Gauthier-Villars, Paris, 1933.
Alexander Ostrowski, Über die analytische Fortsetzung von Taylorschen und Dirichletschen Reihen, Math. Ann. 129 (1955), 1–43 (German). MR 69878, DOI 10.1007/BF01362358
A. F. Leont′ev, On a class of functions defined by series of Dirichlet polynomials, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 4(26), 176–180 (Russian). MR 0027854
A. F. Leont′ev, Ryady polinomov Dirihlet i ih obobščeniya, Izdat. Akad. Nauk SSSR, Moscow, 1951 (Russian). Trudy Mat. Inst. Steklov. no. 39. MR 0055444
Jean-Pierre Kahane, Sur quelques problèmes d’unicité et de prolongement, relatifs aux fonctions approchables par des sommes d’exponentielles, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 39–130 (1955) (French). MR 75350
A. F. Leont′ev, On convergence of a sequence of Dirichlet polynomials, Dokl. Akad. Nauk SSSR (N.S.) 108 (1956), 23–26 (Russian). MR 0080180
A. F. Leont′ev, New proof of a theorem on convergence of a sequence of Dirichlet polynomials, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 3(75), 165–170 (Russian). MR 0089268
A. F. Leont′ev, Properties of sequences of Dirichlet polynomials which are convergent on an interval of the imaginary axis, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 269–328 (Russian). MR 0180809
Laurent Schwartz, Étude des sommes d’exponentielles. 2ième éd, Publications de l’Institut de Mathématique de l’Université de Strasbourg, V. Actualités Sci. Ind., Hermann, Paris, 1959 (French). MR 0106383
Aimée Baillette, Approximation de fonctions par des sommes d’exponentielles, C. R. Acad. Sci. Paris 249 (1959), 2470–2471 (French). MR 117351
Aimée Baillette, Fonctions approchables par des sommes d’exponentielles, J. Analyse Math. 10 (1962/63), 91–115 (French). MR 149168, DOI 10.1007/BF02790304
I. F. Krasičkov, Convergence of Dirichlet polynomials, Sibirsk. Mat. Ž. 7 (1966), 1039–1058 (Russian). MR 0210879
I. F. Krasičkov-Ternovskiĭ, Invariant subspaces of analytic functions. Analytic continuation, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 931–945 (Russian). MR 0367233
A. S. Krivosheev, Analytic continuation of a function from invariant subspaces in convex domains of a complex space, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 2, 75–102 (Russian, with Russian summary); English transl., Izv. Math. 62 (1998), no. 2, 287–312. MR 1623826, DOI 10.1070/im1998v062n02ABEH000174
A. S. Krivosheev, Analytic continuation of functions from invariant subspaces, Dokl. Akad. Nauk 386 (2002), no. 4, 450–452 (Russian). MR 2006037
A. S. Krivosheev, A criterion for the analytic continuation of functional belonging to invariant subspaces in convex domains of the complex plane, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 1, 43–78 (Russian, with Russian summary); English transl., Izv. Math. 68 (2004), no. 1, 43–76. MR 2096937, DOI 10.1070/IM2004v068n01ABEH000466
Christer O. Kiselman, Prolongement des solutions d’une équation aux dérivées partielles à coefficients constants, Bull. Soc. Math. France 97 (1969), 329–356 (French). MR 267259
A. Sebbar, Prolongement des solutions holomorphes de certains opérateurs différentiels d’ordre infini à coefficients constants, Séminaire Pierre Lelong-Henri Skoda (Analyse). Années 1978/79 (French), Lecture Notes in Math., vol. 822, Springer, Berlin, 1980, pp. 199–220 (French). MR 599028
A. Meril and D. C. Struppa, Convolutors in spaces of holomorphic functions, Complex analysis, II (College Park, Md., 1985–86) Lecture Notes in Math., vol. 1276, Springer, Berlin, 1987, pp. 253–275. MR 922324, DOI 10.1007/BFb0078962
A. S. Krivosheev, Indicators of entire functions and the continuation of solutions of a homogeneous convolution equation, Mat. Sb. 184 (1993), no. 8, 81–108 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 79 (1994), no. 2, 401–423. MR 1239760, DOI 10.1070/SM1994v079n02ABEH003507
Ryuichi Ishimura and Yasunori Okada, The existence and the continuation of holomorphic solutions for convolution equations in tube domains, Bull. Soc. Math. France 122 (1994), no. 3, 413–433 (English, with English and French summaries). MR 1294464
Kurt Leichtweiss, Konvexe Mengen, Hochschultext [University Textbooks], Springer-Verlag, Berlin-New York, 1980 (German). MR 586235
L. I. Ronkin, Vvedenie v teoriyu tselykh funktsiĭ mnogikh peremennykh, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0320357
R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
Alexandre Grothendieck, Sur les espaces ($F$) et ($DF$), Summa Brasil. Math. 3 (1954), 57–123 (French). MR 75542
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Pierre Lelong and Lawrence Gruman, Entire functions of several complex variables, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 282, Springer-Verlag, Berlin, 1986. MR 837659, DOI 10.1007/978-3-642-70344-7
A. F. Leont′ev, Tselye funktsii. Ryady èksponent, “Nauka”, Moscow, 1983 (Russian). MR 753827
R. S. Yulmukhametov, Entire functions of several variables with given behavior at infinity, Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), no. 4, 205–224 (Russian, with Russian summary); English transl., Izv. Math. 60 (1996), no. 4, 857–879. MR 1416928, DOI 10.1070/IM1996v060n04ABEH000082
B. Ya. Levin, Distribution of zeros of entire functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0087740