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On Cartwright's theorem


Authors: Natalia Blank and Alexander Ulanovskii
Journal: Proc. Amer. Math. Soc. 144 (2016), 4221-4230
MSC (2010): Primary 30D15, 30D20
DOI: https://doi.org/10.1090/proc/13200
Published electronically: June 10, 2016
MathSciNet review: 3531174
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Abstract: We present a characterization of sets for which Cartwright's theorem holds true. The connection between these sets and sampling sets for entire functions of exponential type is discussed.


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  • [A51] Shmuel Agmon, Functions of exponential type in an angle and singularities of Taylor series, Trans. Amer. Math. Soc. 70 (1951), 492-508. MR 0041222
  • [Be15] Yurii Belov, Complementability of exponential systems, C. R. Math. Acad. Sci. Paris 353 (2015), no. 3, 215-218 (English, with English and French summaries). MR 3306487, https://doi.org/10.1016/j.crma.2014.12.004
  • [Bn48] S. N. Bernšteĭn, The extension of properties of trigonometric polynomials to entire functions of finite degree, Izvestiya Akad. Nauk SSSR. Ser. Mat. 12 (1948), 421-444 (Russian). MR 0027852
  • [Br89] A. Beurling, Balayage of Fourier-Stiltjes Transforms, In: The collected Works of Arne Beurling, vol. 2, Harmonic Analysis, Birkhauser, Boston, 1989.
  • [B40] R. P. Boas Jr., Entire functions bounded on a line, Duke Math. J. 6 (1940), 148-169. MR 0001295
  • [BS42] R. P. Boas Jr. and A. C. Schaeffer, A theorem of Cartwright, Duke Math. J. 9 (1942), 879-883. MR 0007432
  • [B55] R. P. Boas Jr., Growth of analytic functions along a line, J. Analyse Math. 4 (1955), 1-28. MR 0072215
  • [C36] M. L. Cartwright, On certain integral functions of order one, Quart. J. Math. Oxford ser. vol. 7, (1936), 46-55.
  • [DF45] R. J. Duffin and A. C. Schaeffer, Power series with bounded coefficients, Amer. J. Math. 67 (1945), 141-154. MR 0011322
  • [L49] B. Levin, On functions of finite degree, bounded on a sequence of points, Doklady Akad. Nauk SSSR (N.S.) 65 (1949), 265-268 (Russian). MR 0029987
  • [L57] B. Ya. Levin, Generalization of a theorem of Cartwright concerning an entire function of finite degree bounded on a sequence of points, Izv. Akad. Nauk SSSR. Ser. Mat. 21 (1957), 549-558 (Russian). MR 0096798
  • [L96] B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR 1400006
  • [LL93] B. Ya. Levin and V. N. Logvinenko, Classes of functions that are subharmonic in $ {\bf R}^m$ and bounded on certain sets, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), no. Issled. Linein. Oper. Teorii Funktsii. 17, 157-175, 323 (Russian, with English summary); English transl., J. Soviet Math. 63 (1993), no. 2, 202-211. MR 1039578, https://doi.org/10.1007/BF01099311
  • [LM61] H. C. Liu and A. J. Macintyre, Cartwright's theorem on functions bounded at the integers, Proc. Amer. Math. Soc. 12 (1961), 460-462. MR 0125222
  • [L74] V. N. Logvinenko, A certain multidimensional generalization of a theorem of M. Cartwright, Dokl. Akad. Nauk SSSR 219 (1974), 546-549 (Russian). MR 0382694
  • [LF93] V. N. Logvinenko and S. Yu. Favorov, Cartwright-type theorems and real sets of uniqueness for entire functions of exponential type, Mat. Zametki 53 (1993), no. 3, 72-79 (Russian); English transl., Math. Notes 53 (1993), no. 3-4, 294-299. MR 1220186, https://doi.org/10.1007/BF01207716
  • [LN04] V. Logvinenko and N. Nazarova, Bernstein-type theorems and uniqueness theorems, Ukraïn. Mat. Zh. 56 (2004), no. 2, 198-213 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 56 (2004), no. 2, 244-263. MR 2060804, https://doi.org/10.1023/B:UKMA.0000036099.14798.f9
  • [M57] Paul Malliavin, Sur la croissance radiale d'une fonction méromorphe, Illinois J. Math. 1 (1957), 259-296 (French). MR 0089260
  • [PM10] Mishko Mitkovski and Alexei Poltoratski, Pólya sequences, Toeplitz kernels and gap theorems, Adv. Math. 224 (2010), no. 3, 1057-1070. MR 2628803, https://doi.org/10.1016/j.aim.2009.12.014
  • [N70] Rolf Nevanlinna, Analytic functions, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. MR 0279280
  • [OU15] Alexander Olevskii and Alexander Ulanovskii, On irregular sampling in Bernstein spaces, C. R. Math. Acad. Sci. Paris 353 (2015), no. 1, 47-50 (English, with English and French summaries). MR 3285146, https://doi.org/10.1016/j.crma.2014.10.018

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Additional Information

Natalia Blank
Affiliation: Department of Mathematics, Stavanger University, 4036 Stavanger, Norway
Email: natalia.blank@uis.no

Alexander Ulanovskii
Affiliation: Department of Mathematics, Stavanger University, 4036 Stavanger, Norway
Email: alexander.ulanovskii@uis.no

DOI: https://doi.org/10.1090/proc/13200
Keywords: Cartwright's theorem, Beurling's sampling theorem, sampling set
Received by editor(s): August 31, 2015
Published electronically: June 10, 2016
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society

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