Abstract: An obvious property of an arbitrary nonzero smooth antiperiodic function is that its derivative has no corresponding period. In other words, if \(r\) is a fixed positive number, \(f(x+r)+f(x-r)=0\) and \(f'(x+r)-f'(x-r)=0\) on the real axis, then \(f=0\). This fact admits non-trivial generalizations to multidimensional spaces. One general method for such generalizations is the following Brown-Schreiber-Taylor theorem on spectral analysis: any non-zero subspace \(\mathcal{U}\) in \(C(\mathbb{R}^n)\) invariant under all motions of \(\mathbb{R}^n\) contains for some \(\lambda\in \mathbb{C}\), the radial function \((\lambda|x|)^{1-\frac{n}{2}}J_{\frac{n}{2}-1}(\lambda|x|)\), where \(J_{\nu}\) is the Bessel function of the first kind of order \(\nu\). In particular, if a function \(f\in C^1(\mathbb{R}^n)\) and its normal derivative have zero integrals over all spheres of fixed radius \(r\) in \(\mathbb{R}^n\), then \(f=0\). In terms of convolution, this means that the operator \(\mathcal{P}f =(f\ast \Delta \chi_r, f\ast \sigma_r)\), \(f\in C(\mathbb{R}^n)\), is injective, where \(\Delta\) is the Laplace operator, \(\chi_{r}\) is the indicator of the ball \(B_r=\{x\in\mathbb{R}^n: |x| < r\}\), \(\sigma_{r}\) is the surface delta function centered on the sphere \(S_r= \{x\in\mathbb{R}^n: |x|=r\}\). In this paper, we study the inversion problem for the operator \(\mathcal{P}\) on the class of distributions. A new formula for reconstruction a distribution \(f\in \mathcal{D}'(\mathbb{R}^n)\) from known convolutions \(f\ast \Delta \chi_r\) and \(f\ast \sigma_r\) is obtained. The paper uses the methods of harmonic analysis, as well as the theory of entire and special functions. The key step in the proof of the main result is the expansion of the Dirac delta function in terms of a system of radial distributions supported in \(\overline{B}_r\), biorthogonal to some system of spherical functions. A similar approach can be used to invert other convolution operators with radial distributions in \(\mathcal{E}'(\mathbb{R}^n)\).
Keywords: radial distributions, mean periodicity, Pompeiu transform, inversion formulas
For citation: Volchkova, N. P. and Volchkov, Vit. V. Inversion of a Convolution Operator Associated with Spherical Means, Vladikavkaz Math. J. , 2023, vol. 25, no. 3, pp.59-75 (in Russian). DOI 10.46698/z5526-4462-9472-g
1. Minkowski, H. Uber die Korper konstanter Breite,
Gesammelte Abhandlungen, 1911, vol. 2, pp. 277-279. (in German).
2. Funk, P. Uber Flachen mit Lauter Geschlossenen Geodatishen Linien,
Mathematische Annalen, 1913, vol. 74, pp. 278-300. DOI: 10.1007/BF01456044.
3. Radon, J. Uber die Bestimmung von Funktionen durch ihre Integralwerte
langs gewisser Mannigfaltigkeiten, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math.-Nat. Kl., 1917, vol. 69, pp. 262-277.
4. Berenstein, C. A. and Struppa, D. C. Complex Analysis and Convolution Equations,
Several Complex Variables, V, Encyclopedia of Mathematical Sciences, vol. 54, New York,
Springer-Verlag, 1993, pp. 1-108. DOI: 10.1007/978-3-642-58011-6_-1.
5. Zalcman, L. A Bibliographic Survey of the Pompeiu Problem, Approximation
by Solutions of Partial Differential Equations, Dordrecht, Kluwer Academic Publishers, 1992,
pp. 185-194. DOI: 10.1007/978-94-011-2436-2_-17.
6. Zalcman, L. Supplementary Bibliography to "A Bibliographic Survey of the Pompeiu
Problem", Contemporary Mathematics, Radon Transform and Tomography, 2001, vol. 278,
pp. 69-74. DOI: 10.1090/conm/278/04595.
7. Volchkov, V. V. Integral Geometry and Convolution Equations, Dordrecht,
Kluwer Academic Publishers, 2003, 454 p. DOI: 10.1007/978-94-010-0023-9.
8. Volchkov, V. V. and Volchkov, Vit. V. Harmonic Analysis of Mean Periodic
Functions on Symmetric Spaces and the Heisenberg Group, London, Springer, 2009, 672 p. DOI:
10.1007/978-1-84882-533-8.
9. Volchkov, V. V. and Volchkov, Vit. V. Offbeat Integral Geometry on Symmetric
Spaces, Basel, Birkhauser, 2013, 592 p. DOI: 10.1007/978-3-0348-0572-8.
10. Brown, L., Schreiber, B. M. and Taylor, B. A. Spectral Synthesis and the
Pompeiu Problem, Annales de l'Institut Fourier, Grenoble, 1973, vol. 23, no. 3, pp. 125-154. DOI:
10.5802/aif.474.
11. Ikromov, I. A. Recovering a Function from its Spherical Means, Russian
Mathematical Surveys, 1987, vol. 42, no. 5, pp. 169-170. DOI: 10.1070/RM1987v042n05ABEH001476.
12. Berenstein, C. A., Gay, R. and Yger, A. Inversion of the Local Pompeiu Transform,
Journal d'Analyse Mathematique, 1990, vol. 54, no. 1, pp. 259-287. DOI: 10.1007/bf02796152.
13. Helgason, S. Integral Geometry and Radon Transforms, New York, Springer,
2010, 301 p. DOI: 10.1007/978-1-4419-6055-9.
14. Volchkova, N. P. and Volchkov, Vit. V. Deconvolution Problem for Indicators of
Segments, Mathematical Notes of NEFU, 2019, vol. 26, no. 3, pp. 3-14. DOI:
10.25587/SVFU.2019.47.12.001. (in Russian).
15. Hormander, L. The Analysis of Linear Partial Differential Operators,
Vol. I., New York, Springer-Verlag, 2003, 440 p. DOI: 10.1007/978-3-642-61497-2.
16. Helgason, S. Groups and Geometric Analysis, New York, Academic Press,
1984, 667 p.
17. Erd'elyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. Higher
Transcendental Functions (Bateman Manuscript Project), Vol. II, New York, McGraw-Hill, 1953,
302 p.
18. El Harchaoui, M. Inversion de la Transformation de Pomp\'eiu Locale Dans les
Espaces Hyperboliques R'Eel et Complexe (Cas de deux boules), Journal d'Analyse Mathematique, 1995,
vol. 67, no. 1, pp. 1-37. DOI: 10.1007/BF02787785.
19. Berkani, M., El Harchaoui, M. and Gay, R. Inversion de la Transformation de
Pompeiu locale Dans l'espace Hyperbolique Quaternique - Cas des deux boules, Complex Variables, 2000, vol. 43, no. 1, pp. 29-57.
DOI: 10.1080/17476930008815300.
20. Volchkov, Vit. V. and Volchkova, N. P. Inversion of the Local Pompeiu Transform on
the Quaternion Hyperbolic Space, Doklady Mathematics, 2001, vol. 64, no. 1, pp. 90-93.
21. Volchkov, Vit. V. and Volchkova, N. P. Inversion Theorems for the Local Pompeiu
Transformation in the Quaternion Hyperbolic Space, St. Petersburg Mathematical Journal, 2004, vol. 15,
no. 5, pp. 753-771. DOI: 10.1090/S1061-0022-04-00830-1.
22. Volchkov, Vit. V. On Functions with Given Spherical Means on Symmetric
Spaces, Journal of Mathematical Sciences, 2011, vol. 175, no. 4, pp. 402-412. DOI: 10.1007/s10958-011-0354-2.
23. Volchkov, V. V. and Volchkov, Vit. V. Inversion of the Local Pompeiu
Transformation on Riemannian Symmetric Spaces of Rank One, Journal of Mathematical Sciences, 2011, vol. 179,
no. 2, pp. 328-343. DOI: 10.1007/s10958-011-0597-y.
24. Volchkov, V. V. and Volchkov, Vit. V. Spherical Means on Two-Point Homogeneous
Spaces and Applications, Izvestiya: Mathematics, 2013, vol. 77, no. 2, pp. 223-252.
DOI: 10.1070/IM2013v077n02ABEH002634.
25. Rubin, B. Reconstruction of Functions on the Sphere from Their Integrals Over
Hyperplane Sections, Analysis and Mathematical Physics, 2019, vol. 9, no. 4, pp. 1627-1664.
DOI: 10.1007/s13324-019-00290-1.
26. Salman, Y. Recovering Functions Defined on the Unit Sphere by Integration on a
Special Family of Sub-Spheres, Analysis and Mathematical Physics, 2017, vol. 7, no. 2, pp. 165-185.
DOI: 10.1007/s13324-016-0135-7.
27. Hielscher, R. and Quellmalz, M. Reconstructing an Function on the Sphere
from its Means Along Vertical Slices, Inverse Problems and Imaging, 2016, vol. 10, no. 3,
pp. 711-739. DOI: 10.3934/ipi.2016018.
28. Vladimirov, V. S. and Zharinov, V. V. Uravneniya matematicheskoj fiziki
[Equations of Mathematical Pphysics], Moscow, Fizmatlit, 2008, 400 p. (in Russian).
29. Levin, B. Ya. Raspredelenie kornej celyh funkcij [Distribution of
Roots of Entire Functions], Moscow, URSS, 2022, 632 p. (in Russian).
30. Ilyin, V. A., Sadovnichiy, V. A. and Sendov, Bl. Kh. Matematicheskij analiz
[Mathematical Analysis], Vol. II, Moscow, Yurayt-Izdat, 2013, 357 p. (in Russian).
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
