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Subsequences of zeros for Bernstein spaces and the completeness of systems of exponentials in spaces of functions on an interval

Authors: B. N. Khabibullin, G. R. Talipova and F. B. Khabibullin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 2.
Journal: St. Petersburg Math. J. 26 (2015), 319-340
MSC (2010): Primary 30B50; Secondary 42C30
Published electronically: February 3, 2015
MathSciNet review: 3242038
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \sigma >0$. The symbol $ B_\sigma ^\infty $ denotes the space of all entire functions of exponential type not exceeding $ \sigma $ that are bounded on the real axis. Various exact descriptions of uniqueness sequences for the Bernstein spaces $ B_\sigma ^\infty $ are given in terms of $ \sigma $ and the Poisson and Hilbert transformations. These descriptions lead to completeness criteria for systems of exponentials (up to one or two members) in various classical function spaces on an interval (closed or open) of length $ d$.

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  • 1. B. N. Khabibullin, Completeness of exponential systems and uniqueness sets, RIC Bash. Gos. Univ., Ufa, 2012. (Russian)
  • 2. 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
  • 3. Norman Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, v. 26, American Mathematical Society, New York, 1940. MR 0003208
  • 4. Laurent Schwartz, Approximation d’un fonction quelconque par des sommes d’exponentielles imaginaires, Ann. Fac. Sci. Univ. Toulouse (4) 6 (1943), 111–176 (French). MR 0015553
  • 5. Raymond M. Redheffer, Completeness of sets of complex exponentials, Advances in Math. 24 (1977), no. 1, 1–62. MR 0447542
  • 6. Victor Havin and Burglind Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780
  • 7. A. M. Sedletskii, Fourier transforms and approximations, Analytical Methods and Special Functions, vol. 4, Gordon and Breach Science Publishers, Amsterdam, 2000. Translated from the Russian by E. V. Pankratiev. MR 1935577
  • 8. A. M. Sedletskiĭ, Analytic Fourier transforms and exponential approximations. I, Sovrem. Mat. Fundam. Napravl. 5 (2003), 3–152 (electronic) (Russian); English transl., J. Math. Sci. (N. Y.) 129 (2005), no. 6, 4251–4408. MR 2120868, 10.1007/s10958-005-0349-y
  • 9. A. M. Sedletskiĭ, Analytic Fourier transforms and exponential approximations. II, Sovrem. Mat. Fundam. Napravl. 6 (2003), 3–162 (electronic) (Russian); English transl., J. Math. Sci. (N. Y.) 130 (2005), no. 6, 5083–5254. MR 2120869, 10.1007/s10958-005-0397-3
  • 10. N. Makarov and A. Poltoratski, Meromorphic inner functions, Toeplitz kernels and the uncertainty principle, Perspectives in analysis, Math. Phys. Stud., vol. 27, Springer, Berlin, 2005, pp. 185–252. MR 2215727, 10.1007/3-540-30434-7_10
  • 11. Anton Baranov, Completeness and Riesz bases of reproducing kernels in model subspaces, Int. Math. Res. Not. , posted on (2006), Art. ID 81530, 34. MR 2264717, 10.1155/IMRN/2006/81530
  • 12. -, Model subspaces of the Hardy space (Bernstein inequalities systems of reproduction kernels, Beurling-Malliavin type theorems), Doctoral Thesis, St. Petersburg, 2011. (Russian)
  • 13. Frederick W. King, Hilbert transforms. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 124, Cambridge University Press, Cambridge, 2009. MR 2542214
  • 14. Frederick W. King, Hilbert transforms. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 125, Cambridge University Press, Cambridge, 2009. MR 2542215
  • 15. J. N. Pandey, The Hilbert transform of Schwartz distributions and applications, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1996. A Wiley-Interscience Publication. MR 1363489
  • 16. Laurent Schwartz, Analyse mathématique. II, Hermann, Paris, 1967 (French). MR 0226973
  • 17. B. N. Khabibullin, Applications in complex analysis of dual representation for functionals on vector lattices, Mathematical Forum, vol. 4, Researches on Math. Analysis Differential Equations and their Applications, Vladikavkaz, 2010, pp. 102-116. (Russian)
  • 18. B. N. Khabibullin, Sets of uniqueness in spaces of entire functions of one variable, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 1101–1123 (Russian); English transl., Math. USSR-Izv. 39 (1992), no. 2, 1063–1084. MR 1149889
  • 19. S. A. Grigoryan, Generalized analytic functions, Uspekhi Mat. Nauk 49 (1994), no. 2(296), 3–42 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 2, 1–40. MR 1283134, 10.1070/RM1994v049n02ABEH002202
  • 20. B. N. Khabibullin, Dual representation of superlinear functionals and its applications in the theory of functions. II, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 5, 167–190 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 5, 1017–1039. MR 1874358, 10.1070/IM2001v065n05ABEH000361
  • 21. -, The distribution of zeros of entire function and the balayage, Doctoral Thesis, Khar'kov, 1993. (Russian)
  • 22. Pierre Blanchet, On removable singularities of subharmonic and plurisubharmonic functions, Complex Variables Theory Appl. 26 (1995), no. 4, 311–322. MR 1315864
  • 23. B. N. Khabibullin, Completeness of systems of entire functions in spaces of holomorphic functions, Mat. Zametki 66 (1999), no. 4, 603–616 (Russian, with Russian summary); English transl., Math. Notes 66 (1999), no. 3-4, 495–506 (2000). MR 1747088, 10.1007/BF02679100
  • 24. Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766
  • 25. John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
  • 26. M. Brelot, Éléments de la théorie classique du potentiel, Les Cours de Sorbonne. 3e cycle, Centre de Documentation Universitaire, Paris, 1959 (French). MR 0106366
  • 27. Paul Koosis, The logarithmic integral. II, Cambridge Studies in Advanced Mathematics, vol. 21, Cambridge University Press, Cambridge, 1992. MR 1195788
  • 28. B. N. Khabibullin, Nonconstructive proofs of the Beurling-Malliavin theorem on the radius of completeness, and nonuniqueness theorems for entire functions, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 4, 125–148 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 45 (1995), no. 1, 125–149. MR 1307059, 10.1070/IM1995v045n01ABEH001622
  • 29. Paul Koosis, Leçons sur le théorème de Beurling et Malliavin, Université de Montréal, Les Publications CRM, Montreal, QC, 1996 (French). MR 1430571

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Additional Information

B. N. Khabibullin
Affiliation: Bashkir State University, ul. Zaki Validi 32, Ufa 450074, Bashkortostan, Russia

G. R. Talipova
Affiliation: Bashkir State University, ul. Zaki Validi 32, Ufa 450074, Bashkortostan, Russia

F. B. Khabibullin
Affiliation: Bashkir State University, ul. Zaki Validi 32, Ufa 450074, Bashkortostan, Russia

Keywords: Entire function, Bernstein space, sequence of uniqueness, completeness of exponentials, Poisson integral, Hilbert transformation
Received by editor(s): February 4, 2012
Published electronically: February 3, 2015
Additional Notes: Supported by RFBR (grant no. 13-01-00030a) and by the Federal targeted program “Scientific and pedagogical staff of innovative Russia” (contract no. 14.B37.21.0358)
Article copyright: © Copyright 2015 American Mathematical Society