Geodesic
A basis in an invariant subspace of analytic functions
Sbornik. Mathematics, Tome 204 (2013) no. 12, pp. 1745-1796 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The existence problem for a basis in a differentiation-invariant subspace of analytic functions defined in a bounded convex domain in the complex plane is investigated. Conditions are found for the solvability of a certain special interpolation problem in the space of entire functions of exponential type with conjugate diagrams lying in a fixed convex domain. These underlie sufficient conditions for the existence of a basis in the invariant subspace. This basis consists of linear combinations of eigenfunctions and associated functions of the differentiation operator, whose exponents are combined into relatively small clusters. Necessary conditions for the existence of a basis are also found. Under a natural constraint on the number of points in the groups, these coincide with the sufficient conditions. That is, a criterion is found under this constraint that a basis constructed from relatively small clusters exists in an invariant subspace of analytic functions in a bounded convex domain in the complex plane. Bibliography: 25 titles.
Keywords: exponential polynomial, invariant subspace, basis.
Mots-clés : interpolation 
@article{SM_2013_204_12_a3,
     author = {A. S. Krivosheev and O. A. Krivosheeva},
     title = {A basis in an invariant subspace of analytic functions},
     journal = {Sbornik. Mathematics},
     pages = {1745--1796},
     year = {2013},
     volume = {204},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_12_a3/}
}
TY  - JOUR
AU  - A. S. Krivosheev
AU  - O. A. Krivosheeva
TI  - A basis in an invariant subspace of analytic functions
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 1745
EP  - 1796
VL  - 204
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_12_a3/
LA  - en
ID  - SM_2013_204_12_a3
ER  - 
%0 Journal Article
%A A. S. Krivosheev
%A O. A. Krivosheeva
%T A basis in an invariant subspace of analytic functions
%J Sbornik. Mathematics
%D 2013
%P 1745-1796
%V 204
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2013_204_12_a3/
%G en
%F SM_2013_204_12_a3
A. S. Krivosheev; O. A. Krivosheeva. A basis in an invariant subspace of analytic functions. Sbornik. Mathematics, Tome 204 (2013) no. 12, pp. 1745-1796. http://geodesic.mathdoc.fr/item/SM_2013_204_12_a3/

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

[2] I. F. Krasichkov-Ternovskij, “A homogeneous equation of convolution type on convex domains”, Soviet Math. Dokl., 12 (1971), 396–398 | MR | Zbl

[3] A. A. Gol'dberg, B. Ya. Levin, I. V. Ostrovkij, “Entire and meromorphic functions”, Complex Analysis I, Encyclopaedia Math. Sci., 85, Springer-Verlag, Berlin, 1997, 1–193 | MR | MR | Zbl | Zbl

[4] A. S. Krivosheev, “The fundamental principle for invariant subspaces in convex domains”, Izv. Math., 68:2 (2004), 291–353 | DOI | DOI | MR | Zbl

[5] O. A. Krivosheeva, A. S. Krivosheev, “Fundamentalnyi printsip dlya invariantnykh podprostranstv”, Ufimsk. matem. zhurn., 2:4 (2010), 58–73

[6] A. F. Leontev, Tselye funktsii. Ryady eksponent, Nauka, M., 1983 | MR | Zbl

[7] D. G. Dickson, “Infinite order differential equations”, Proc. Amer. Math. Soc., 15:4 (1964), 638–641 | DOI | MR | Zbl

[8] A. F. Leontev, “O predstavlenii funktsii posledovatelnostyami polinomov Dirikhle”, Matem. sb., 70(112):1 (1966), 132–144 | MR | Zbl

[9] R. Meise, “Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals”, J. Reine Angew. Math., 363 (1985), 59–95 | DOI | MR | Zbl

[10] V. V. Napalkov, “A basis in the space of solutions of a convolution equation”, Math. Notes, 43:1 (1988), 27–33 | DOI | MR | Zbl | Zbl

[11] A. S. Krivosheev, “The Schauder basis in the solution space of a homogeneous convolution equation”, Math. Notes, 57:1 (1995), 41–50 | DOI | MR | Zbl

[12] A. S. Krivosheev, “Pochti eksponentsialnyi bazis”, Ufimsk. matem. zhurn., 2:1 (2010), 87–96

[13] A. S. Krivosheev, “Bazisy «po otnositelno malym gruppam»”, Ufimsk. matem. zhurn., 2:2 (2010), 67–89 | Zbl

[14] A. S. Krivosheev, “Pochti eksponentsialnaya posledovatelnost eksponentsialnykh mnogochlenov”, Ufimsk. matem. zhurn., 4:1 (2012), 88–106

[15] A. Grothendieck, “Sur les espaces ($F$) et ($DF$)”, Summa Brasil. Math., 3 (1954), 57–123 | MR | Zbl

[16] O. A. Krivosheyeva, “The convergence domain for series of exponential monomials”, Ufa Math. Journal, 3:2 (2011), 42–55

[17] B. Ja. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, RI, 1964 | MR | MR | Zbl | Zbl

[18] I. F. Krasichkov-Ternovskii, “A geometric lemma useful in the theory of entire functions and Levinson-type theorems”, Math. Notes, 24:4 (1978), 784–792 | DOI | MR | Zbl | Zbl

[19] P. Lelong, L. Gruman, Entire functions of several complex variables, Grundlehren Math. Wiss., 282, Springer-Verlag, Berlin, 1986 | MR | MR | Zbl | Zbl

[20] O. A. Krivosheeva, “Singular points of the sum of a series of exponential monomials on the boundary of the convergence domain”, St. Petersburg Math. J., 23:2 (2012), 321–350 | DOI | MR | Zbl

[21] L. Hörmander, The analysis of linear partial differential operators, v. 1, Grundlehren Math. Wiss., 256, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[22] L. I. Ronkin, Introduction to the theory of entire functions of several variables, Amer. Math. Soc., Providence, RI, 1974 | MR | MR | Zbl | Zbl

[23] R. S. Yulmukhametov, “Approksimatsiya subgarmonicheskikh funktsii”, Anal. Math., 11:3 (1985), 257–282 | DOI | MR | Zbl

[24] A. S. Krivosheev, V. V. Napalkov, “Complex analysis and convolution operators”, Russian Math. Surveys, 47:6 (1992), 1–56 | DOI | MR | Zbl

[25] A. F. Leontev, Ryady eksponent, Nauka, M., 1976 | MR | Zbl