Recovering the laplacian from centered means on balls and spheres of fixed radius

N. P. Volchkova ; Vit. V. Volchkov

Problemy analiza, Tome 12 (2023) no. 1, pp. 96-117.

Résumé

Various issues related to restrictions on radii in mean-value formulas are well-known in the theory of harmonic functions. In particular, using the Brown-Schreiber-Taylor theorem on spectral synthesis for motion-invariant subspaces in $C(\mathbb{R}^n)$, one can obtain the following strengthening of the classical mean-value theorem for harmonic functions: if a continuous function on $\mathbb{R}^n$ satisfies the mean-value equations for all balls and spheres of a fixed radius $r$, then it is harmonic on $\mathbb{R}^n$. In connection with this result, the following problem arises: recover the Laplacian from the deviation of a function from its average values on balls and spheres of a fixed radius. The aim of this work is to solve this problem. The article uses 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 expansion of the Dirac delta function in terms of a system of radial distributions supported in a fixed ball, biorthogonal to some system of spherical functions. A similar approach can be used to invert a number of convolution operators with compactly supported radial distributions.

Détail

Keywords: harmonic functions, one-radius theorems
Mots-clés : radial distributions, Fourier-Bessel transform.

Bibliographie

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

[2] Berenstein C. A., Struppa D. C., “Complex analysis and convolution equations”, Several Complex Variables V, Chap. 1, Encyclopedia of Math. Sciences, 54, 1993, 1–108 | DOI

[3] 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 (2000), 29–57 | DOI | MR | Zbl

[4] Blaschke W., “Ein Mittelwersatz und eine kennzeichnende Eigenschaft des logaritmischen Potentials”, Ber. Ver. Sächs Akad. Wiss. Leipzig, 68 (1916), 3–7

[5] Brown L., Schreiber B. M., Taylor B. A., “Spectral synthesis and the Pompeiu problem”, Ann. Inst. Fourier, Grenoble, 23:3 (1973), 125–154 | DOI | MR | Zbl

[6] Casey S. D., Walnut D. F., “Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms”, Siam Review, 36:4 (1994), 537–577 | DOI | MR | Zbl

[7] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, v. II, McGraw-Hill, New York, 1953, 316 pp. | MR

[8] Flatto L., “The converse of Gauss theorem for harmonic functions”, J. Different. Equat., 1 (1965), 483–490 | DOI | MR | Zbl

[9] Helgason S., Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Amer. Math. Soc., New York, 2000, 667 pp. | MR | Zbl

[10] Hörmander L., The Analysis of Linear Partial Differential Operators I, Springer-Verlag, New York, 2003, 440 pp. | DOI | MR | Zbl

[11] Ikromov I.A., “Recovering a function from its spherical means”, Russian Mathematical Surveys, 42:5 (1987), 169–170 | DOI | MR | Zbl

[12] Kuznetsov N., “Mean value properties of harmonic functions and related topics (a survey)”, J. Math. Sci., 242:2 (2019), 177–199 | DOI | MR | Zbl

[13] Netuka I., Veselý J., “Mean value property and harmonic functions”, Classical and Modern Potential Theory and Applications, NATO ASI Series, 430, Springer, Dordrecht, 1994, 359–398 | MR

[14] Nicolesco M., Les Fonctions Polyharmoniques, Hermann, Paris, 1936

[15] Pizzetti P., “Sulla media dei valori che una funzione del punti dello spazio assume alla superficie di una sfera”, Rend. Lincei, Ser. 5, 18 (1909), 182–185

[16] Poritsky H., “Generalizations of the Gauss law of the spherical mean”, Trans. Amer. Math. Soc., 43 (1938), 199–225 | DOI | MR | Zbl

[17] Priwaloff J., “Sur les fonctions harmoniques”, Mat. Sb., 32:3 (1925), 464–471

[18] Timan A. F., Trofimov V. N., Introduction to the Theory of Harmonic Functions, Nauka, M., 1968, 208 pp. (in Russian) | MR

[19] Volchkov V. V., “Theorems on two radii on bounded domains of Euclidean spaces”, Differ. Equ., 30:10 (1994), 1587–1592 | MR | Zbl

[20] Volchkov V. V., “New two-radii theorems in the theory of harmonic functions”, Ivz. Math., 44:1 (1995), 181–192 | DOI | MR

[21] Volchkov V. V., Volchkov Vit. V., Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group, Springer-Verlag, London, 2009, 672 pp. | DOI | MR | Zbl

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

[23] 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

[24] Volchkov V. V., Volchkov Vit. V., Offbeat Integral Geometry on Symmetric Spaces, Birkhäuser, Basel, 2013, 592 pp. | DOI | MR | Zbl

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

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