Abstract
We study functions on a sphere with a pricked point having zero integrals with a given weight over all admissible “hemispheres”. We find a condition under which the point is a removable set for such a class of functions. We show that this condition cannot be dropped or substantially weakened.
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References
Shabat, B. V. Introduction to Complex Analysis (Nauka, Moscow, 1985), Vol. 2 [in Russian].
Martio, O., Ryazanov, V., Srebro, U., Yakubov, E. Moduli in Modern Mapping Theory (Springer, New York, 2009).
Axler, S., Bourdon, P., Ramey, W. Harmonic Function Theory (Springer-Verlag, New York, 1992).
Markushevich, A. I. Selected Chapters of the Theory of Analytic Functions (Nauka, Moscow, 1976) [in Russian].
Trokhimchuk, Yu. Yu. Continuous Mappings and Monogeneity Conditions (Fizmatgiz, Moscow, 1963) [in Russian].
Helgason, S. Integral Geometry and Radon Transforms (Springer, New York, 2010).
Volchkov, V. V. Integral Geometry and Convolution Equations (Kluwer Academic Publ., Dordrecht, 2003).
Volchkov V. V., Volchkov, Vit. V. Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group (Springer, London, 2009).
Volchkov, V. V., Volchkov, Vit. V. Offbeat Integral Geometry on Symmetric Spaces (Birkhäuser, Basel, 2013).
Quinto, E. T. “Pompeiu Transforms on Geodesic Spheres in Real Analytic Manifolds”, Israel J. Math. 84, 353–363 (1983).
Quinto, E. T. “Radon Transforms on Curves in the Plane”, in E. T. Quinto, M. Cheney, P. Kuchment (Eds.) Tomography, Impedance Imaging, and Integral Geometry (South Hadley, MA), Lectures in Appl. Math. 30, 231–244 (1994).
Zhou, Y. “Two Radius Support Theorem for the Sphere Transform”, J. Math. Anal. Appl. 254, 120–137 (2001).
Volchkov, V. V. “Solution of the Support Problem for Several Function Classes”, Sb.Math. 188 (9), 1279–1294 (1997).
Stein, I., Weiss, G. Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, NJ1971; Mir, Moscow, 1974).
Funk, P. “Über eine geometrische Anwendung der Abelschen Integralgleichung”, Math. Annal. 77, 129–135 (1916).
Rubin, B. “Inversion and Characterization of the Hemispherical Transform”, J. D’AnalyseMath. 77, 105–128 (1999).
Campi S. “On the Reconstruction of a Star-Shaped Body From Its ‘Half-Volumes’”, J. Austral.Math. Soc. (Ser. A) 37, 243–257 (1984).
Batemen, H., Erdélyi, A. Higher Transcendental Functions (McGraw-Hill, New York–Toronto–London, 1953; Nauka, Moscow, 1973 (2nd ed.)), Vol.1.
Volchkov, Vit. V., Savost’yanova, I. M. “On a Kernel of the Hemispherical Fourier Transform and Its Local Analogs”, Ukr.Matem. Vestnik 10, No. 4, 575–594 (2013) [in Russian].
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Original Russian Text © Vit.V. Volchkov, N.P. Volchkova, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 8, pp. 17–26.
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Volchkov, V.V., Volchkova, N.P. The extension problem for functions with zero weighted spherical means. Russ Math. 61, 13–21 (2017). https://doi.org/10.3103/S1066369X17080023
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DOI: https://doi.org/10.3103/S1066369X17080023