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

Khabibullin, B.; Talipova, G.; Khabibullin, F.

doi:10.1090/S1061-0022-2015-01340-X

Subsequences of zeros for Bernstein spaces and the completeness of systems of exponentials in spaces of functions on an interval
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by B. N. Khabibullin, G. R. Talipova and F. B. Khabibullin
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 26 (2015), 319-340

DOI: https://doi.org/10.1090/S1061-0022-2015-01340-X

Published electronically: February 3, 2015

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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 $d$.

References

B. N. Khabibullin, Completeness of exponential systems and uniqueness sets, RIC Bash. Gos. Univ., Ufa, 2012. (Russian)
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
Norman Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940. MR 0003208
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 15553
Raymond M. Redheffer, Completeness of sets of complex exponentials, Advances in Math. 24 (1977), no. 1, 1–62. MR 447542, DOI 10.1016/S0001-8708(77)80002-9
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, DOI 10.1007/978-3-642-78377-7
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
A. M. Sedletskiĭ, Analytic Fourier transforms and exponential approximations. I, Sovrem. Mat. Fundam. Napravl. 5 (2003), 3–152 (Russian); English transl., J. Math. Sci. (N.Y.) 129 (2005), no. 6, 4251–4408. MR 2120868, DOI 10.1007/s10958-005-0349-y
A. M. Sedletskiĭ, Analytic Fourier transforms and exponential approximations. II, Sovrem. Mat. Fundam. Napravl. 6 (2003), 3–162 (Russian); English transl., J. Math. Sci. (N.Y.) 130 (2005), no. 6, 5083–5254. MR 2120869, DOI 10.1007/s10958-005-0397-3
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, DOI 10.1007/3-540-30434-7_{1}0
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, DOI 10.1155/IMRN/2006/81530
—, Model subspaces of the Hardy space (Bernstein inequalities systems of reproduction kernels, Beurling–Malliavin type theorems), Doctoral Thesis, St. Petersburg, 2011. (Russian)
Frederick W. King, Hilbert transforms. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 124, Cambridge University Press, Cambridge, 2009. MR 2542214
Frederick W. King, Hilbert transforms. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 125, Cambridge University Press, Cambridge, 2009. MR 2542215
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
Laurent Schwartz, Analyse mathématique. II, Hermann, Paris, 1967 (French). MR 0226973
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)
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, DOI 10.1070/IM1992v039n02ABEH002237
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, DOI 10.1070/RM1994v049n02ABEH002202
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, DOI 10.1070/IM2001v065n05ABEH000361
—, The distribution of zeros of entire function and the balayage, Doctoral Thesis, Khar′kov, 1993. (Russian)
Pierre Blanchet, On removable singularities of subharmonic and plurisubharmonic functions, Complex Variables Theory Appl. 26 (1995), no. 4, 311–322. MR 1315864, DOI 10.1080/17476939508814792
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, DOI 10.1007/BF02679100
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
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
M. Brelot, Éléments de la théorie classique du potentiel, “Les Cours de Sorbonne”, vol. 3, Centre de Documentation Universitaire, Paris, 1959 (French). MR 0106366
Paul Koosis, The logarithmic integral. II, Cambridge Studies in Advanced Mathematics, vol. 21, Cambridge University Press, Cambridge, 1992. MR 1195788, DOI 10.1017/CBO9780511566202
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, DOI 10.1070/IM1995v045n01ABEH001622
Paul Koosis, Leçons sur le théorème de Beurling et Malliavin, Université de Montréal, Les Publications CRM, Montreal, QC, 1996 (French). MR 1430571