On Cartwright’s theorem
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- by Natalia Blank and Alexander Ulanovskii PDF
- Proc. Amer. Math. Soc. 144 (2016), 4221-4230 Request permission
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.References
- Shmuel Agmon, Functions of exponential type in an angle and singularities of Taylor series, Trans. Amer. Math. Soc. 70 (1951), 492–508. MR 41222, DOI 10.1090/S0002-9947-1951-0041222-5
- 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, DOI 10.1016/j.crma.2014.12.004
- 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
- A. Beurling, Balayage of Fourier–Stiltjes Transforms, In: The collected Works of Arne Beurling, vol. 2, Harmonic Analysis, Birkhauser, Boston, 1989.
- R. P. Boas Jr., Entire functions bounded on a line, Duke Math. J. 6 (1940), 148–169. MR 1295
- R. P. Boas Jr. and A. C. Schaeffer, A theorem of Cartwright, Duke Math. J. 9 (1942), 879–883. MR 7432
- R. P. Boas Jr., Growth of analytic functions along a line, J. Analyse Math. 4 (1955), 1–28. MR 72215, DOI 10.1007/BF02787715
- M. L. Cartwright, On certain integral functions of order one, Quart. J. Math. Oxford ser. vol. 7, (1936), 46–55.
- R. J. Duffin and A. C. Schaeffer, Power series with bounded coefficients, Amer. J. Math. 67 (1945), 141–154. MR 11322, DOI 10.2307/2371922
- 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
- 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
- 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, DOI 10.1090/mmono/150
- B. Ya. Levin and V. N. Logvinenko, Classes of functions that are subharmonic in $\textbf {R}^m$ and bounded on certain sets, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), no. Issled. Lineĭn. Oper. Teorii Funktsiĭ. 17, 157–175, 323 (Russian, with English summary); English transl., J. Soviet Math. 63 (1993), no. 2, 202–211. MR 1039578, DOI 10.1007/BF01099311
- 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 125222, DOI 10.1090/S0002-9939-1961-0125222-4
- V. N. Logvinenko, A certain multidimensional generalization of a theorem of M. Cartwright, Dokl. Akad. Nauk SSSR 219 (1974), 546–549 (Russian). MR 0382694
- 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, DOI 10.1007/BF01207716
- 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, DOI 10.1023/B:UKMA.0000036099.14798.f9
- Paul Malliavin, Sur la croissance radiale d’une fonction méromorphe, Illinois J. Math. 1 (1957), 259–296 (French). MR 89260
- Mishko Mitkovski and Alexei Poltoratski, Pólya sequences, Toeplitz kernels and gap theorems, Adv. Math. 224 (2010), no. 3, 1057–1070. MR 2628803, DOI 10.1016/j.aim.2009.12.014
- Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 0279280
- 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, DOI 10.1016/j.crma.2014.10.018
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
- MR Author ID: 194862
- Email: alexander.ulanovskii@uis.no
- Received by editor(s): August 31, 2015
- Published electronically: June 10, 2016
- Communicated by: Alexander Iosevich
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4221-4230
- MSC (2010): Primary 30D15, 30D20
- DOI: https://doi.org/10.1090/proc/13200
- MathSciNet review: 3531174