On the zeros of mean-periodic functions with respect to the Bessel convolution
https://doi.org/10.22405/2226-8383-2024-25-5-57-73
Abstract
In paper, we study uniqueness sets for solutions to the Bessel convolution equation 𝑓𝛼 ⋆𝑔 = 0, 𝛼 ∈ (−1/2,+∞). It is shown, in particular, that if 𝑔 = 𝜒𝑟 is an indicator function of the segment [−𝑟, 𝑟], and an even function 𝑓 ∈ 𝐶(R) satisfies the equation 𝑓𝛼 ⋆𝜒𝑟 = 0
and is zero on (𝑟 − 𝜀, 𝑟) or (𝑟, 𝑟 + 𝜀) for some 𝜀 > 0, then 𝑓 = 0 on (𝑟 − 𝜀, 𝑟 + 𝜀). In this case, the interval
of zeros (𝑟 − 𝜀, 𝑟 + 𝜀), cannot generally be extended. It has been established that a similar phenomenon occurs for solutions of the equation 𝑓𝛼 ⋆𝛿𝑟 = 0, where 𝛿𝑟 is an even measure that maps an even continuous function 𝜙 on R to a number 𝜙(𝑟). Applications of these results to uniqueness theorems for convergent sequences of linear combinations of Bessel functions, the zero set theorem for solutions of the Cauchy problem of the generalized Euler-Poisson-Darboux
equation and the closure theorem of generalized shifts are found.
About the Authors
Vitaly Vladimirovich Volchkov
Donetsk National University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Gleb Vitalyevich Krasnoschekikh
Donetsk National University
Russian Federation
Russian Federation
postgraduate student
References
1. Delsarte, J. 1935, “Les fonctions ‘moyenne-p´eriodiques’ ”, J. Math. Pures Appl., vol. 14, no. 9, pp. 403–453.
2. Schwartz, L. 1947, “Theorie g´en´erale des functions moyenne-periodiqu˝e ”, Annals of Mathematics, vol. 48, pp. 857–928.
3. Kahane, J.P. 1957, “Sur les fonctions moyenne–p´eriodiques born´ees ”, Ann. Inst. Fourier
4. (Grenoble), vol. 7, pp. 293–314.
5. Nikol’skii, N.K. 1976, “Invariant subspaces in the theory of operators and theory of functions”, Journal of Soviet Mathematics, vol. 7, no. 2, pp. 129–249. doi: 10.1007/BF01247397/
6. Berenstein, K., Struppa, D. 1989, “Complex analysis and convolution equations”, Itogi Nauki i Tekhniki, vol. 54, pp. 5–111. doi: 10.1007/978-3-652-58011-6/
7. Titchmarsh, E.C. 1937, “Introduction to the theory of Fourier integrals”, Clarendon Press,
8. Oxford, 390 p.
9. John, F. 1955, “Plane waves and spherical means applied to partial differential equations”, Dover Publications, Mineola, NY, 172 p. doi: 10.1007/978-1-4613-9453-2/
10. Lyubich, Yu.I. 1956, “On a class of integral equations”, Mat. Sb. (N.S.), vol. 38(80), no. 2,
11. pp. 183-202.
12. Leont’ev, A. F. 1965, “Properties of sequences of Dirichlet polynomials converging on an interval of the imaginary axis”, Izv. Akad. Nauk SSSR Ser. Mat., vol. 29, no. 2, pp. 269–328.
13. Smith, J.D. 1972, “Harmonic analysis of scalar and vector fields in R𝑛”, Proc. Cambridge Philos. Soc., vol 72, pp. 403–416. doi: 10.1017/S0305004100047241/
14. Leont’ev, A. F. 1980, “Sequences of exponential polynomials”, Nauka, Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury, Moscow, 384 p.
15. Volchkov, V.V. 1995, “A definitive version of the local two-radii theorem”, Sbornik: Mathematics, vol. 186, no. 6, pp. 783-802. doi: 10.1070/SM1995v186n06ABEH000043/
16. Volchkov, V.V. 2003, “Integral Geometry and Convolution Equations”, Kluwer Academic
17. Publishers, Dordrecht, 454 p. doi: 10.1007/978-94-010-0023-9/
18. Volchkov, V.V., Volchkov, Vit.V. 2009, “Harmonic Analysis of Mean Periodic Functions on
19. Symmetric Spaces and the Heisenberg Group”, Springer, London, 671 p. doi: 10.1007/978-1-
20. -533-8/
21. Zaraisky, D.A. 2012, “A uniqueness theorem for functions with zero integrals over balls”, Tr. Inst. Prikl. Mat. Mekh., vol. 25, pp. 77–83.
22. Zaraisky, D.A. 2019, “A class of mean-periodic functions which are uniquely determined by their restrictions to the ‘period’ ”, Tr. Inst. Prikl. Mat. Mekh., vol. 33, pp. 38–41
23. Zaraisky, D.A. 2019, “The uniqueness theorem for solutions of the convolution equation with convolution kernel”, Materialy‘ IV Mezhd. nauch. konf. “Doneczkie chteniya 2019: obrazovanie, nauka, innovacii, kul‘tura i vy‘zovy‘ sovremennosti” ”, (Proc. 4-th Int. Sci. Conf. “Donetsk Readings 2019: education, science, innovation, culture and modern problems”). Donetsk, 2019, pp. 127–128.
24. Quinto, E.T. 1993, “Pompeiu transforms on geodesic spheres in real analytic manifolds”, Israel J. Math., vol. 84, pp. 353–363. doi: 10.1007/BF02760947/
25. Lyubich, Yu.I. 1984, “Uniqueness theorem for mean-periodic functions”, Journal of Soviet
26. Mathematics, vol. 26, no. 5, p. 2206. doi: 10.1007/BF01221539/
27. Kargaev, P.P. 1985, “Zeros of mean-periodic functions”, Mathematical Notes, vol. 37, no. 3, pp. 181–183. doi: 10.1007/BF01158736/
28. Zaraisky, D.A. 2006, “Refinement of the uniqueness theorem for solutions of the convolution equation”, Tr. Inst. Prikl. Mat. Mekh., vol. 12, pp. 69–75.
29. Volchkov, V.V., Volchkov, Vit.V. 2010, “Optimization problems related to the John uniqueness theorem”, St. Petersburg Mathematical Journal, vol. 21, no. 5, pp. 705–729. doi: 10.1090/S1061-0022-2010-01113-0/
30. Volchkov, V.V., Volchkov, Vit.V. 2013, “Offbeat Integral Geometry on Symmetric Spaces”,
31. Birkh¨auser, Basel, 592 p. doi: 10.1007/978-3-0348-0572-8/
32. Volchkov, V.V., Volchkov, Vit.V. 2013, “Spherical means on two-point homogeneous spaces and applications”, Izv. Math., vol. 77, no. 2, pp. 223–252. doi: 10.4213/im7956/
33. Volchkov, V.V. 1999, “Injectivity sets for the Radon transform over a sphere”, Izv. Math., vol. 63, no. 3, pp. 481–493. doi: 10.4213/im248/
34. Volchkov, V.V. 1996, “Extremal cases of the Pompeiu problem”, Math. Notes, vol. 59, no. 5, pp. 482–489. doi: 10.4213/mzm1761/
35. Zaraisky, D.A. 2020, “A new uniqueness theorem for one-dimensional convolution equation”, Tr. Inst. Prikl. Mat. Mekh., vol. 34, pp. 63–67.
36. Selmi, B., Nessibi, M.M. 2007, “A Local Two Radii Theorem on the Ch´ebli-Trim`eche
37. Hypergroup”, Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 163–190. doi: 10.1016/j.jmaa.2006.06.061/
38. Volchkov, Vit.V., Krasnoschekikh, G.V. 2024, “A Refinement of the Two-Radius Theorem
39. on the Bessel-Kingman Hypergroup”, Mathematical Notes, vol. 116, no. 2, pp. 223–237.
40. doi: 10.4213/mzm14184/
41. Volchkov, V.V. 1992, “Uniqueness theorems for multiple lacunary trigonometric series”, Math. Notes, vol. 51, no. 6, pp. 550–552. doi: 10.1007/BF01263296/
42. Paley, R., Wiener, N. 1936, “Fourier transforms in the complex domain”, (American Math. Soc. Colloq. Publications Vol. XIX)VIII+184 Seiten. Amer. Mathem. Society, New York, vol. 44, pp. A8–A9. doi: 10.1007/BF01699343/
43. Levinson, N. 1940, “Gap and density theorems”, Amer. Math. Soc., Providence, 243 p.
44. Shiskina, E.L., Sitnik, S.M. 2019, “Method of Transmutations for Differential Equations with Bessel Operators”, Fizmatlit, Moscow, 224 p.
45. Bateman, H. Erd´elyi, A. 1973, 1974, “Higher transcendental functions. Vol. 1–2.”, Nauka,
46. Moscow, 294, 296 p.
Review
For citations:
Volchkov V.V., Krasnoschekikh G.V. On the zeros of mean-periodic functions with respect to the Bessel convolution. Chebyshevskii Sbornik. 2024;25(5):57-73. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-5-57-73
Views:
320
Email this article 