Geodesic
On rate of decreasing of extremal function in Carleman class
Ufa mathematical journal, Tome 15 (2023) no. 2, pp. 31-41 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the issues related with Levinson-Sjöberg-Wolf type theorems in the complex analysis and, in particular, we discuss a famous question posed in 70s by E.M. Dyn'kin on an effective bound for majorant of the growth of an analytic function in the vicinity of the set of singular points and another close problem on the rate of decaying of an extremal function in a non-quasianalytic Carleman class in the vicinity of the point, at which all the derivatives of the functions from this class vanish. Exact asymptotic estimates for the best majorant for the growth in the vicinity of the singularities were found by V. Matsaev and M. Sodin in 2002. Some bounds, both from above and below, for an extremal function in the Carleman class were obtained by A.M. Gaisin in 2018 but they turned out to be not very close to exact values of this function. In the present paper we obtain sharp two-sided estimates for the extremal function.
Keywords: Levinson-Sjöberg type theorem, extremal function, regular sequence, associated weight.
Mots-clés : non-quasianalytic Carleman class 
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R. A. Gaisin. On rate of decreasing of extremal function in Carleman class. Ufa mathematical journal, Tome 15 (2023) no. 2, pp. 31-41. http://geodesic.mathdoc.fr/item/UFA_2023_15_2_a3/

[1] N. Levinson, Gap and density theorems, Amer. Math. Soc., New-York, 1940 | MR | Zbl

[2] V.P. Gurarii, “On the Levinson theorem on normal families of analytic functions”, Zapis. Nauchn. Semin. POMI, 19, 1970, 215–220 (in Russian) | Zbl

[3] V. Matsaev, Theorems on uniqueness, completeness and compactness, related with the classical quasianalyticity, PhD thesis, Kharkov, 1964 (in Russian)

[4] E.M. Dyn'kin, “The growth of an analytic function near its set of singular points”, Zapis. Nauchn. Semin. POMI, 30, 1972, 158–160 (in Russian) | Zbl

[5] E.M. Dyn'kin, “Functions with given estimate for $\frac{\partial f} {\partial \overline{z}}$ and N. Levinson's theorem”, Math. USSR-Sb., 18:2 (1972), 181–189 | DOI | Zbl

[6] E.M. Dyn'kin, “The pseudoanalytic extension”, J. Anal. Math., 60:1 (1993), 45–70 | DOI | MR | Zbl

[7] A.M. Gaisin, I.G. Kinzyabulatov, “A Levinson-Sjöberg type theorem. Applications”, Sb. Math., 199:7 (2008), 985–1007 | DOI | DOI | MR | Zbl

[8] N. Sjoberg, “Sur les minorantes subharmoniques d'une function donee”, Comp. rendus IX Congres des Math. (Scandinaves, Helsingfors), 1939, 309–319 | Zbl

[9] T. Carleman, “Extension d'un theoreme de Liouville”, Acta Math., 48:3-4 (1926), 363–366 | DOI | MR

[10] F. Wolf, “On majorants of subharmonic and analytic functions”, Bull. Amer. Math. Soc., 48:12 (1942), 925–932 | DOI | MR | Zbl

[11] P. Koosis, The logarithmic integral, v. I, Cambridge Univ. Press, Cambridge, 1988 | MR | Zbl

[12] Y. Domar, “On the existence of a largest subharmonic minorant of a given function”, Ark. Mat., 3:5 (1958), 429–440 | DOI | MR | Zbl

[13] A.M. Gaisin, “Extremal problems in nonquasianalytic Carleman classes. Applications”, Sb. Math., 209:7 (2018), 958–984 | DOI | DOI | MR | Zbl

[14] V. Matsaev, M. Sodin, “Asymptotics of Fourier and Laplace transforms in weighted spaces of analytic functions”, St.-Petersburg Math. J., 14:4 (2003), 615–640 | MR | Zbl

[15] N. Nikol'skii, Yngve Domar's forty years in harmonic analysis, Uppsala Univ, Uppsala, 1995, 45–78 | MR | Zbl

[16] T. Bang, “The theory of metric spaces applied to infinitely differentiable functions”, Math. Scand., 1 (1953), 137–152 | DOI | MR

[17] S. Mandelbrojt, Séries adhérentes. Régularisation des suites. Applications, Gauthier-Villars, Paris, 1952 (in French) | MR