The problem of finding a function by its ball means values

N. P. Volchkova ; Vit. V. Volchkov

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 63-80.

Résumé

A classical property of a non-constant $2r$-periodic function on the real axis is that it has no period incommensurable with $r$. One of the multidimensional analogues of this statement is the following well-known theorem of L. Zalcman on two radii: for the existence of a nonzero locally summable function $f:\mathbb{R}^n\to \mathbb{C}$ with nonzero integrals over all balls of radii $r_1$ and $r_2$ in $\mathbb{R}^n$ it is necessary and sufficient that $r_1/r_2\in E_n$, where $E_n$ is the set of all possible ratios of positive zeros of the Bessel function $J_{n/2}$. The condition $r_1/r_2\notin E_n$is equivalent to the equality $\mathcal{Z}_{+}\big(\widetilde{\chi}_{r_1}\big)\cap\mathcal{Z}_{+}\big(\widetilde{\chi}_{r_2}\big)=\varnothing$, where $\chi_{r}$ is the indicator of the ball $B_r=\{x\in\mathbb{R}^n: |x|$, $\widetilde{\chi}_{r}$ is the spherical transform (Fourier-Bessel transform) of the indicator $\chi_{r}$, $\mathcal{Z}_{+}(\widetilde{\chi}_{r})$ is the set of all positive zeros of even entire function $\widetilde{\chi}_{r}$. In terms of convolutions, L. Zalcman's theorem means that the operator $$\mathcal{P}f=(f\ast \chi_{r_1}, f\ast \chi_{r_2}), f\in L^{1,\mathrm{loc}}(\mathbb{R}^n) $$ is injective if and only if $r_1/r_2\notin E_n$. In this paper, a new formula for the inversion of the operator $\mathcal{P}$ is found under the condition $r_1/r_2\notin E_n$. The result obtained significantly simplifies the previously known procedures for recovering a function $f$ from given ball means values $f\ast \chi_{r_1}$ и $f\ast \chi_{r_2}$. The proofs use the methods of harmonic analysis, as well as the theory of entire and special functions.

Détail

Keywords: mean periodic functions, radial distributions, two-radii theorem, inversion formulas.

Bibliographie

[1] Pompéiu D., “Sur certains systèmes d'équations linéaires et sur une propriété intégrale de fonctions de plusieurs variables”, C. R. Acad. Sci. Paris, 188 (1929), 1138–1139

[2] Pompéiu D., “Sur une propriété intégrale de fonctions de deux variables réeles”, Bull. Sci. Acad. Royale Belgique (5), 15 (1929), 265–269 | Zbl

[3] Chakalov L., “Sur un problème de D. Pompeiu”, Annuaire [Godišnik] Univ. Sofia Fac. Phys.-Math., Livre 1, 40 (1944), 1–14 | MR | Zbl

[4] Berenstein, C.A., Struppa, D.C., “Complex analysis and convolution equations”, Several complex variables. $\mathrm{V}$: Complex analysis in partial differential equations and mathematical physics, Encyclopaedia of Mathematical Sciences, 54, 1993, 1–108 | DOI

[5] Zalcman L., “A bibliographic survey of the Pompeiu problem”, Approximation by solutions of partial differential equations, 365 (1992), 185–194 | DOI | MR | Zbl

[6] Zalcman L., “Supplementary bibliography to “A bibliographic survey of the Pompeiu problem””, Radon Transform and Tomography, Contemp. Math., 278, 2001, 69–74 | DOI | MR | Zbl

[7] Volchkov V. V., Integral Geometry and Convolution Equations, Kluwer Academic Publishers, Dordrecht, 2003 | DOI | MR | Zbl

[8] Volchkov V. V., Volchkov Vit. V., Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group, Springer, London, 2009 | DOI | MR | Zbl

[9] Volchkov V. V., Volchkov Vit. V., Offbeat Integral Geometry on Symmetric Spaces, Birkhäuser, Basel, 2013 | DOI | MR | Zbl

[10] Delsarte J., “Note sur une propriété nouvelle des fonctions harmoniques”, C. R. Acad. Sci. Paris Sér. A–B, 246 (1958), 1358–1360 | MR | Zbl

[11] Zalcman L., “Analyticity and the Pompeiu problem”, Arch. Rat. Anal. Mech., 47:3 (1972), 237–254 | DOI | MR | Zbl

[12] Smith J. D., “Harmonic analysis of scalar and vector fields in $\mathbb R^n$”, Proc. Cambridge Philos. Soc., 72:3 (1972), 403–416 | DOI | MR | Zbl

[13] Zalcman L., “Offbeat integral geometry”, Amer. Math. Monthly, 87:3 (1980), 161–175 | DOI | MR | Zbl

[14] Berenstein C. A., Taylor B. A., Yger A., “Sur quelques formules explicites de déconvolution”, J. Optics (Paris), 14:2 (1983), 75–82 | DOI | MR

[15] Berenstein C. A., Yger A., “Le problème de la déconvolution”, J. Funct. Anal., 54:2 (1983), 113–160 | DOI | MR | Zbl

[16] Volchkov, V. V., “A definitive version of the local two-radii theorem”, Sb. Math., 186:6 (1995), 783–802 | DOI | MR | Zbl

[17] Berenstein C. A., Yger A., “Analytic Bezout identities”, Adv. Appl. Math., 10:1 (1989), 51–74 | DOI | MR | Zbl

[18] Berenstein C. A., Gay R., Yger A., “Inversion of the local Pompeiu transform”, J. Analyse Math., 54:1 (1990), 259–287 | DOI | MR | Zbl

[19] Helgason S., Geometric Analysis on Symmetric spaces, Amer. Math. Soc., Providence, Rhode Island, 2008 http://books.google.com/books?vid=ISBN978-1-4704-1266-1 | Zbl

[20] Volchkov, V. V., Volchkov, Vit. V., “Convolution equations in many-dimensional domains and on the Heisenberg reduced group”, Sb. Math., 199:8 (2008), 1139–1168 | DOI | MR | Zbl

[21] Hörmander L., The Analysis of Linear Partial Differential Operators, v. I, Springer-Verlag, New York, 2003 | DOI | MR | Zbl

[22] Helgason S., Groups and Geometric Analysis, Academic Press, New York, 1984 | MR | Zbl

[23] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Higher Transcendental Functions, v. II, McGraw-Hill, New York, 1953 https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738

[24] Volchkova, N. P., Volchkov, Vit. V., “Deconvolution problem for indicators of segments”, Math. Notes NEFU, 26:3 (2019), 3–14 | DOI | MR

[25] El Harchaoui M., “Inversion de la transformation de Pompéiu locale dans les espaces hyperboliques réel et complexe (Cas de deux boules)”, J. Anal. Math., 67:1 (1995), 1–37 | DOI | MR | Zbl

[26] Berkani M., El Harchaoui M., Gay R., “Inversion de la transformation de Pompéiu locale dans l'espace hyperbolique quaternique - Cas des deux boules”, J. Complex Variables, 43:1 (2000), 29–57 | DOI | MR | Zbl

[27] Volchkov, Vit. V., Volchkova, N. P., “Inversion of the local Pompeiu transform on the quaternion hyperbolic space”, Dokl. Math., 64:1 (2001), 90–93 | MR | Zbl | Zbl

[28] Volchkov, Vit. V., Volchkova, N. P., “nversion theorems for the local Pompeiu transformation in the quaternion hyperbolic space”, St. Petersburg Math. J., 15:5 (2004), 753–771 | DOI | MR | Zbl

[29] Volchkov Vit. V., “On functions with given spherical means on symmetric spaces”, J. Math. Sci., 175:4 (2011), 402–412 | DOI | MR | Zbl

[30] Volchkov V. V., Volchkov Vit. V., “Inversion of the local Pompeiu transformation on Riemannian symmetric spaces of rank one”, J. Math. Sci., 2011:2, 328–343 | DOI | MR | Zbl

[31] Volchkov, V. V., Volchkov, Vit. V., “Spherical means on two-point homogeneous spaces and applications”, Ivz. Math., 77:2 (2013), 223–252 | DOI | MR | Zbl

[32] Rubin B., “Reconstruction of functions on the sphere from their integrals over hyperplane sections”, Anal. Math. Phys., 9:4 (2019), 1627–1664 | DOI | MR | Zbl

[33] Salman Y., “Recovering functions defined on the unit sphere by integration on a special family of sub-spheres”, Anal. Math. Phys., 7:2 (2017), 165–185 | DOI | MR | Zbl

[34] Hielscher R., Quellmalz M., “Reconstructing an function on the sphere from its means along vertical slices”, Inverse Probl. Imaging, 10:3 (2016), 711–739 | DOI | MR | Zbl

[35] Vladimirov, V. S., Zharinov, V. V., Equations of mathematical physics, Fizmatlit, M., 2008

[36] Levin, B. Ya., Distribution of roots of entire functions, URSS, M. \, 2022

[37] Ilyin, V. A., Sadovnichiy, V. A., Sendov, Bl. Kh., Mathematical analysis, v. II, Yurayt-Izdat., M., 2013 | MR