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Gaĭsin, Akhtyar Magazovich; Gaĭsina, Galiya Akhtyarovna

Estimate for growth and decay of functions in Macintyre-Evgrafov kind theorems. (Russian. English summary) Zbl 1456.30004

Ufim. Mat. Zh. 9, No. 3, 27-37 (2017); translation in Ufa Math. J. 9, No. 3, 26-36 (2017).
Summary: In the paper we obtain two results on the behavior of Dirichlet series on the real axis.
The first of them concerns the lower bound for the sum of the Dirichlet series on the system of segments \([\alpha,\,\alpha+\delta]\). Here the parameters \(\alpha > 0\), \(\delta > 0\) are such that \(\alpha \uparrow + \infty\), \(\delta \downarrow 0\). The needed asymptotic estimates is established by means of a method based on some inequalities for extremal functions in the appropriate non-quasi-analytic Carleman class. This approach turns out to be more effective than the known traditional ways for obtaining similar estimates.
The second result specifies essentially the known theorem by M. A. Evgrafov on existence of a bounded on \(\mathbb{R}\) Dirichlet series. According to Macintyre, the sum of this series tends to zero on \(\mathbb{R} \). We prove a spectific estimate for the decay rate of the function in an Macintyre-Evgrafov type example.

Cited in 1 Document

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
30D10 Representations of entire functions of one complex variable by series and integrals

Keywords:

Dirichlet series; entire functions
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References:

[1] G. Pólya, “Untersuchungen über Lücken und Singularitäten von Potenzreihen”, Math. Z., 29:1 (1929), 549-640 · JFM 55.0186.02 · doi:10.1007/BF01180553
[2] M. N. Sheremeta, “A property of entire functions with real taylor coefficients”, Math. Notes, 18:3 (1975), 823-827 · Zbl 0318.30026 · doi:10.1007/BF01095439
[3] M. M. Sheremeta, M. V. Zabolotskii, “Some open problems in theory of functions of a complex variable”, Matem. Studii, 3 (1994), 117-119 (Problem section)
[4] M.M. Sheremeta, “Five open problems in the theory of entire functions”, Matem. Studii, 6 (1996), 157-159 (Problem section)
[5] A. M. Gaisin, “Solution of the Polya problem”, Sb. Math., 193:6 (2002), 825-845 · Zbl 1067.30051 · doi:10.1070/SM2002v193n06ABEH000659
[6] A. M. Gaisin, “Estimates of the growth and decrease on curves of an entire function of infinite order”, Sb. Math., 194:8 (2003), 1167-1194 · Zbl 1062.30027 · doi:10.1070/SM2003v194n08ABEH000761
[7] A. M. Gaisin, “On a theorem of Hayman”, Siber. Math. J., 39:3 (1998), 431-445 · Zbl 0905.30024 · doi:10.1007/BF02673898
[8] A. M. Gaisin, Zh. G. Rakhmatullina, “An estimate for the sum of a Dirichlet series in terms of the minimum of its modulus on a vertical line segment”, Sb. Math., 202:12 (2011), 1741-1773 · Zbl 1253.30009 · doi:10.1070/SM2011v202n12ABEH004206
[9] M. A. Evgrafov, “On a uniqueness theorem for Dirichlet series”, Uspekhi Matem. Nauk, 17:3 (1962), 169-175 (in Russian) · Zbl 0115.06201
[10] A.J. Macintyre, “Asymptotic paths of integral functions with gap power series”, Proc. London Math. Soc., 2:3 (1952), 286-296 · Zbl 0048.05502 · doi:10.1112/plms/s3-2.1.286
[11] N. N. Yusupova, Asymptotics of the Dirichlet series of prescribed growth, PhD thesis, Inst. Math., Ufa, 2009 · Zbl 1176.30008
[12] A. M. Gaisin, “Properties of series of exponentials whose exponents satisfy to a condition of Levinson type”, Sborn. Math., 197:6 (2006), 813-833 · Zbl 1151.30021 · doi:10.4213/sm1569
[13] A. M. Gaisin, “Levinson”s condition in the theory of entire functions: equivalent statements”, Math. Notes, 83:3 (2008), 317-326 · Zbl 1158.30015 · doi:10.4213/mzm3850
[14] A. F. Leontiev, Exponential series, Nauka, M., 1976 (in Russian) · Zbl 0433.30002
[15] A. M. Gaisin, “Dirichlet series with real coefficients that are unbounded on the positive half-axis”, Sb. Math., 198:6 (2007), 793-815 · Zbl 1162.30004 · doi:10.1070/SM2007v198n06ABEH003861
[16] A. M. Gaisin, Borel-Nevalinna type theorems. Applications, Editing Publishing Center, Bashkir State Univ., Ufa, 2010 (in Russian)
[17] A. M. Gaisin, “Strong incompleteness of a system of exponentials, and Macintyre”s problem”, Math. USSR-Sb., 73:2 (1992), 305-318 · Zbl 0735.30005
[18] A. M. Gaisin, “An estimate for a Dirichlet series whose exponents are zeros of an entire function with irregular behavior”, Russ. Acad. Sci. Sb. Math., 81:1 (1995), 163-183 · Zbl 0832.30002
[19] I. I. Hirschman, D. V. Widder, The convolution transform, Princeton Univ. Press, Princeton, 1955 · Zbl 0065.09301
[20] A. F. Leont’ev, Sequences of exponential polynomials, Nauka, M., 1980 (in Russian) · Zbl 0478.30005
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