Basis in an invariant space of entire functions

Krivosheev, A.; Krivosheeva, O.

doi:10.1090/spmj/1387

Basis in an invariant space of entire functions
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by A. S. Krivosheev and O. A. Krivosheeva
Translated by: S. Kislyakov

St. Petersburg Math. J. 27 (2016), 273-316

DOI: https://doi.org/10.1090/spmj/1387

Published electronically: January 29, 2016

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Abstract:

The existence of a basis is studied in a space of entire functions invariant under the differentiation operator. It is proved that every such space possesses a basis consisting of linear combinations of generalized eigenvectors. These linear combinations are formed within groups of exponents of arbitrarily small relative diameter. A complete description of the way to split the exponents into groups is obtained. Also, a criterion is found for the existence of a basis constructed by groups of zero relative diameter (so-called relatively small groups). In this connection a new criterion is obtained for the finiteness of the lower indicator of an entire function of exponential type.

References

A. F. Leont′ev, Tselye funktsii. Ryady èksponent, “Nauka”, Moscow, 1983 (Russian). MR 753827
V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, “Nauka”, Moscow, 1982 (Russian). MR 678923
A. F. Leont′ev, Posledovatel′nosti polinomov iz èksponent, “Nauka”, Moscow, 1980 (Russian). MR 577300
Laurent Schwartz, Théorie générale des fonctions moyenne-périodiques, Ann. of Math. (2) 48 (1947), 857–929 (French). MR 23948, DOI 10.2307/1969386
A. O. Gel′fond, Linear differential equations of infinite order with constant coefficients and asymptotic periods of entire functions, Trudy Mat. Inst. Steklov. 38 (1951), 42–67 (Russian). MR 0047776
A. S. Krivosheev, The fundamental principle for invariant subspaces in convex domains, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 2, 71–136 (Russian, with Russian summary); English transl., Izv. Math. 68 (2004), no. 2, 291–353. MR 2058001, DOI 10.1070/IM2004v068n02ABEH000476
I. F. Krasičkov, Lower bound for entire functions of finite order, Sibirsk. Mat. Ž. 6 (1965), 840–861 (Russian). MR 0193236
A. S. Krivosheev and O. A. Krivosheeva, A basis in an invariant subspace of analytic functions, Mat. Sb. 204 (2013), no. 12, 49–104 (Russian, with Russian summary); English transl., Sb. Math. 204 (2013), no. 11-12, 1745–1796. MR 3185085, DOI 10.1070/sm2013v204n12abeh004359
B. Ya. Levin, Distribution of zeros of entire functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0087740
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
V. S. Azarin, The indicators of an entire function, and the regularity of the growth of the Fourier coefficients of the logarithm of its modulus, Funkcional. Anal. i Priložen. 9 (1974), no. 1, 47–48 (Russian). MR 0367202
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
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
—, Basis by “relatively small clusters”, Ufim. Mat. Zh. 2 (2010), no. 2, 67–69. (Russian)
—, An almost exponential sequence of exponential polynomials, Ufim. Mat. Zh. 4 (2012), no. 1, 88–106; English transl., Ufa Math. J. 4 (2012), no. 1, 82–100.
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
L. I. Ronkin, Vvedenie v teoriyu tselykh funktsiĭ mnogikh peremennykh, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0320357
O. A. Krivosheeva, Singular points of the sum of series of exponential monomials on the boundary of the convergence domain, Algebra i Analiz 23 (2011), no. 2, 162–205 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 2, 321–350. MR 2841675, DOI 10.1090/S1061-0022-2012-01199-4
A. S. Krivosheev, An almost exponential basis, Ufim. Mat. Zh. 2 (2010), no. 1, 87–96. (Russian)
O. A. Krivosheeva, Convergence domain for series of exponential polynomials, Ufim. Mat. Zh. 5 (2013), no. 4, 84–90; English transl., Ufa Math. J. 5 (2013), no. 4, 82–87.
Alexandre Grothendieck, Sur les espaces ($F$) et ($DF$), Summa Brasil. Math. 3 (1954), 57–123 (French). MR 75542