Geodesic
A Levinson-Sjöberg type theorem. Applications
Sbornik. Mathematics, Tome 199 (2008) no. 7, pp. 985-1007 Cet article a éte moissonné depuis la source Math-Net.Ru

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A generalization of the well-known Levinson-Sjöberg theorem is obtained for a family of analytic functions $f$ that have estimates of the form $|f(z)|\le M(\operatorname{dist}(z,\gamma))$ outside an arc $\gamma$, where $M$ is a decreasing function on $(0,\infty)$ that is unbounded in a neighbourhood of the origin. Applications to questions of quasianalyticity for Carleman classes are indicated as well as to the completeness of a system of exponentials on arcs, to analytic continuation and to representation by Dirichlet series. Bibliography: 24 titles. 
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A. M. Gaisin; I. G. Kinzyabulatov. A Levinson-Sjöberg type theorem. Applications. Sbornik. Mathematics, Tome 199 (2008) no. 7, pp. 985-1007. http://geodesic.mathdoc.fr/item/SM_2008_199_7_a2/

[1] N. Levinson, Gap and density theorems, Amer. Math. Soc. Colloq. Publ., 26, Amer. Math. Soc., New York, 1940 | MR | Zbl

[2] N. Sjöberg, “Sur les minorantes sous-harmoniques d'une fonction donnée”, Comp. rendus IX Congreś des Math. Scandinaves Helsingfors, 1939, 309–319 | Zbl

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

[4] V. P. Gurarij, “On Levinson's theorem concerning normal families of analytic functions”, Sem. Math. V. A. Steklov, 19 (1972), 124–127 | MR | Zbl

[5] E. M. Dyn'kin, “Growth of an analytic function near its set of singular points”, J. Math. Sci., 4:4 (1975), 438–440 | DOI | MR | Zbl

[6] R. Nevanlinna, Eindeutige analytische Funktionen, Julius Springer, Berlin, 1936 | MR | Zbl

[7] G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969 | MR | MR | Zbl | Zbl

[8] P. Koosis, The logarithmic integral, vol. I, Cambridge Stud. Adv. Math., 12, Cambridge Univ. Press, Cambridge, 1988 | MR | Zbl

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

[10] S. Mandelbrojt, Séries adhérentes. Régularisation des suites. Applications, Gauthier, Paris, 1952 | MR | MR | Zbl

[11] A. F. Leontev, Posledovatelnosti polinomov iz eksponent, Nauka, M., 1980 | MR | Zbl

[12] J. Korevaar, “Approximation on curves by linear combinations of exponentials”, Approximation theory (Univ. Texas, Austin, TX 1973), Academic Press, New York, 1973, 387–393 | MR | Zbl

[13] Th. Bang, Om quasianalytiske functioner, Univ. of Copenhagen, 1946 | Zbl

[14] R. Zeinstra, “Zeros and regular growth of Laplace transforms along curves”, J. Reine Angew. Math., 424 (1992), 1–15 | MR | Zbl

[15] E. M. Dyn'kin, “Pseudoanalytic extension of smooth functions. The uniform scale”, Trans. Amer. Math. Soc. Ser. 2, 115 (1980), 33–58 | MR | Zbl

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

[17] V. I. Matsaev, Nekotorye teoremy polnoty i kompaktnosti, svyazannye s klasicheskoi kvazianalitichnostyu, Dis. ... dokt. fiz.-matem. nauk, Izd-vo Khark. un-ta, Kharkov, 1964

[18] A. M. Gaisin, “Properties of exponential series with sequence of exponents satisfying a Levinson-type condition”, Sb. Math., 197:6 (2006), 813–833 | DOI

[19] M. Dixon, J. Korevaar, “Nonspanning sets of powers on curves: analyticity theorem”, Duke Math. J., 45:3 (1978), 543–559 | DOI | MR | Zbl

[20] A. M. Gaĭsin, “Strong incompleteness of a system of exponentials, and Macintyre's problem”, Math. USSR-Sb., 73:2 (1992), 305–318 | DOI | MR | Zbl | Zbl

[21] A. Baillette, J. A. Siddiqi, “Non-totalité d'exponentielles sur un arc rectifiable”, C. R. Acad. Sci. Paris Sér. A-B, 289:3 (1979), 177–179 | MR | Zbl

[22] M. A. Evgrafov, “Ob odnoi teoreme edinstvennosti dlya ryadov Dirikhle”, UMN, 17:3 (1962), 169–175 | MR | Zbl

[23] A. E. Fryntov, Operatory, sokhranyayuschie subgarmonichnost, i nekotorye zadachi klasicheskogo kompleksnogo analiza, Dis. ... dokt. fiz.-matem. nauk, FTINT NAN Ukrainy, Kharkov, 1995

[24] B. Chabat, Introduction à l'analyse complexe. Tome 1. Fonctions d'une variable, Moscow, Mir, 1990 | MR | MR | Zbl | Zbl