Abstract
We establish necessary and sufficient conditions for the system of values of a holomorphic vector function to be complete in a Fréchet space. The results obtained are illustrated by several examples.
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REFERENCES
T. Carlemann, “Ñber die Approximation analytischer Function durch lineare Aggregate von vorgegebenen Potenzen,” Arkiv für Math. Astr. och Fysic., 17 (1922), no. 9, 1–30.
A. O. Gel'fond, “On complete systems of analytic functions,” Mat. Sb. [Math. USSR-Sb.], 4(46) (1938), 149–156.
A. I. Markushevich, “A basis in the space of analytic functions,” Mat. Sb. [Math. USSR-Sb.], 17(59) (1945), 211–252.
A. F. Leont'ev, “Completeness of a system of analytic functions, ”Mat. Sb. [Math. USSR-Sb.], 31(73) (1952), 318–413.
A. F. Leont'ev, “Series of Dirichlet polynomials and their genealizations,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.]., 39 (1951), 3–212.
A. F. Leont'ev, Series of Exponentials [in Russian ], Nauka, Moscow, 1976.
A. F. Leont'ev, A Generalization of Series of Exponentials [in Russian ], Nauka, Moscow, 1981.
A. F. Leont'ev, “On sequences of linear aggregates formed from solutions of differential equations,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 22 (1958), no. 2, 201–242.
I. F. Krasichkov, “On completeness of functions of the form \(\frac{{\partial ^{n_k } }}{{\partial h^{n_k } }}F(z;\;h_k )\), ”Mat. Sb. [Math. USSR-Sb.], 56(98) (1962), 147–172.
I. F. Krasichkov, “Completeness in spaces of complex-valued functions described by the behavior of the modulus, ”Mat. Sb. [Math. USSR-Sb.], 68(110) (1965), no. 1, 26–57.
Yu. A. Kaz'min, “On completeness of systems of functions of the form \(\{ f(z + \alpha _n )\}\) and \(\{ f^{(n)} (z)\}\),” Uspekhi Mat. Nauk [Russian Math. Surveys], 12 (1957), no. 2, 151–154.
Yu. A. Kaz'min, “On completeness of a system of analytic functions I,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], (1956), no. 5, 3–12.
V. P. Gromov, “On sequences of linear aggregates of functions of several complex variables,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 180 (1968), no. 4, 778–782.
W. Rudin, Principles of Mathematical Analysis, McGraw Hill, New York, 1964.
V. P. Gromov, “Order and type of a linear operator and series expansion in eigenfunctions,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 228 (1986), no. 1, 27–31.
V. P. Gromov, “Order and type of a linear operator and entire vector-valued functions,” in: Uchen. Zap. Orlovsk. Gos. Univ. [in Russian ], Orel State Univ., Orel, 1999, pp. 6–23.
A. Yu. Popov, “Approximation properties of several types of sequences of entire functions,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 106 (1999), no. 1, 18–20.
V. P. Gromov, “To the problem of cyclic elements of the differentiation operator in spaces of analytic functions,” Mat. Zametki [Math. Note], 33 (1983), no. 4, 529–538.
S. N. Mishin, “Order and type of operators,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 381 (2001), no. 3, 309–312.
S. N. Mishin, “Order and type of operators and sequences of operators acting in locally convex spaces,”in: Uchen. Zap. Orlovsk. Gos. Univ. [in Russian ], Orel State Univ., Orel, 2002. no. 3, pp. 47–99.
V. P. Gromov, “On completeness of systems of arbitrary analytic functions,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 25 (1961), no. 4, 548–556.
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Gromov, V.P. Completeness of the System of Values of a Holomorphic Vector Function in Fréchet Spaces. Mathematical Notes 73, 783–795 (2003). https://doi.org/10.1023/A:1024093528517
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DOI: https://doi.org/10.1023/A:1024093528517