Subharmonic additions to Beurling – Malliavin Theorems. I. On the multiplier
Abstract
The Beurling – Malliavin Theorem on the multiplier and its various versions give several variants of conditions for the function f on the real axis R, under which this function can be multiplied by an entire bounded on R function h of arbitrarily small exponential type > 0 so that the product of fh is bounded on R. We consider a new version for the function f = exp(u − M), where u and M are a pair of subharmonic functions of finite type with finite logarithmic integrals over R.
Keywords
entire function of exponential type,
subharmonic function of finite type,
multiplier,
Cartwright class
Supporting agencies: The work was carried out within the framework of the state task of the Ministry of Science and Higher Education of the Russian Federation (the code of the scientific topic FMRS-2022-0124).
About the Authors
B. N. Khabibullin
Institute of Mathematics with Computing Centre Subdivision of the Ufa Federal Research Centre of Russian Academy of Science
Russian Federation
Russian Federation
Bulat Nurmievich Khabibullin
112 Chernyshevsky str., Ufa 450008
E. G. Kudasheva
Akmulla Bashkir State Pedagogical University
Russian Federation
Russian Federation
Elena Gennadievna Kudasheva
3A Oktyabr’skoy Revolyutsii str., Ufa, 450008
References
1. R.P. Boas, Jr., Entire Functions, Academic Press, N.Y., 1954.
2. B.Ya. Levin, Lectures on entire functions, Transl. Math. Monographs 150, AMS, Providence R.I., 1996.
3. L.A. Rubel (with J.E. Colliander), Entire and Meromorphic Functions, Springer-Verlag, New York–Berlin–Heidelberg, 1996.
4. B.N. Khabibullin, Completeness of exponential systems and uniqueness sets, 4th ed., Bashkir State Univ. Press, Ufa, 2012 [in Russian]. DOI: https://doi.org/10.13140/2.1.4572.7525
5. B.Ja. Levin, Distribution of Zeros of Entire Functions, AMS, 1964.
6. V. Havin, B. J¨oricke, The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, 1994.
7. P. Koosis, The logarithmic integral. I, Cambridge Stud. Adv. Math. 12, Cambridge Univ. Press, Cambridge, 1988.
8. P. Koosis, The logarithmic integral. II, Cambridge Stud. Adv. Math. 21, Cambridge Univ. Press, Cambridge, 1992.
9. P. Koosis, Le¸cons sur le th´eor`eme de Beurling et Malliavin, Les Publications CRM, Univ. Montr´eal, Montr´eal, QC, 1996.
10. J. Mashreghi, F.L. Nazarov and V.P. Havin Beurling–Malliavin multiplier theorem: The seventh proof, St. Petersburg Math. J. 17 (5), 699–744 (2006). DOI: https://doi.org/10.1090/S1061-0022-06-00926-5
11. A. Beurling, P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (3-4), 291–309 (1962). DOI: https://doi.org/10.1007/BF02545792
12. P. Malliavin, On the multiplier theorem for Fourier transforms of measures with compact support, Ark. Mat. 17 (1-2), 69–81 (1979). DOI: https://doi.org/10.1007/BF02385458
13. Ch.O. Kiselman, Order and type as measures of growth for convex or entire functions, Proc. London Math. Soc. 66 (3), 152–186 (1993). DOI: https://doi.org/10.1112/plms/s3-66.1.152
14. B.N. Khabibullin, A.V. Shmeleva Balayage of measures and subharmonic functions to a system of rays. I. The classical case St. Petersburg Math. J. 31 (1), 117–156 (2020). DOI: https://doi.org/10.1090/spmj/1589
15. B.N. Khabibullin, Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic, Sb.: Math. 213 (5), 694–733 (2022). DOI: https://doi.org/10.1070/SM9642
16. L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, CRC Press, 2015.
17. Th. Ransford, Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995. DOI: https://doi.org/10.1017/CBO9780511623776
18. V. Matsaev, M. Sodin, Distribution of Hilbert transforms of measures, Geom. Funct. Anal. 10 (1), 160–184 (2000). DOI: https://doi.org/10.1007/s000390050005
19. T. Yu. Baˇiguskarov, G.R. Talipova, B.N. Khabibullin, Subsequences of zeros for classes of entire functions of exponential type distinguished by growth restrictions, St. Petersburg Math. J. 28 (2), 127–151 (2017). DOI: https://doi.org/10.1090/spmj/1442
20. B.N. Khabibullin, A.V. Shmeleva and Z.F. Abdullina, Balayage of measures and subharmonic functions to a system of rays. II. Balayages of finite genus and growth regularity on a single ray, St. Petersburg Math. J. 32 (1), 155–181 (2021). DOI: https://doi.org/10.1090/spmj/1642
21. A.E. Salimova, B.N. Khabibullin, Growth of subharmonic functions along line and distribution of their Riesz measures, Ufa Math. J. 12 (2), 35–49 (2020). DOI: https://doi.org/10.13108/2020-12-2-35
22. W.K. Hayman, Subharmonic functions. II, Academic Press, London, 1989.
23. B.N. Khabibullin, On the Growth of Entire Functions of Exponential Type near a Straight Line, Math. Notes 70 (4), 560–573 (2001). DOI: https://doi.org/10.1023/A:1012393106283
24. B.N. Khabibullin, The restriction from below of the subharmonic function by the logarithm of the module of entire function, arXiv:2203.12383 (2022) [in Russian]. URL: https://arxiv.org/abs/2203.12383
25. B.N. Khabibullin, Distributions of zeros and masses of entire and subharmonic functions with restrictions on their growth along the strip, Izv. RAN, Ser. Matem. (accepted paper) [in Russian].
26. A. Beurling, P. Malliavin, On the closure of characters and the zeros of entire functions, Acta Math. 118, 79–93 (1967). DOI: https://doi.org/10.1007/BF02392477
27. J.-P. Kahane, Travaux de Beurling et Malliavin, S´eminaire Bourbaki (ann´ee 1961/62, expos´es 223–240, Talk no. 225), (7), 27–39 (1962). URL: http://www.numdam.org/item/SB_1961-1962__7__27_0/
28. R.M. Redheffer, Completeness of sets of complex exponentials, Adv. in Math. 24 (1), 1–62 (1977). DOI: https://doi.org/10.1016/S0001-8708(77)80002-9
29. I.F. Krasichkov-Ternovskii, An interpretation of the Beurling–Malliavin theorem on the radius of completeness, Math. USSR-Sb. 66 (2), 405–429 (1990). DOI: https://doi.org/10.1070/SM1990v066n02ABEH001178
30. B.N. Khabibullin, Nonconstructive proofs of the Beurling–Malliavin theorem on the radius of completeness, and nonuniqueness theorems for entire functions, Russian Acad. Sci. Izv. Math. 45 (1), 125–149 (1995). DOI: https://doi.org/10.1070/IM1995v045n01ABEH001622
31. B.N. Khabibullin, G.R. Talipova, F.B. Khabibullin, Zero subsequences for Bernsteins spaces and the completeness of exponential systems in spaces of functions on an interval, St. Petersburg Math. J. 26 (2), 319–340 (2015). DOI: https://doi.org/10.1090/S1061-0022-2015-01340-X
Review
For citations:
Khabibullin B.N., Kudasheva E.G. Subharmonic additions to Beurling – Malliavin Theorems. I. On the multiplier. Mathematics and Theoretical Computer Science. 2023;1(3):59-76. (In Russ.)
Views:
133
Email this article 