We prove the existence of nontrivial functions in ℝn, n ≥ 2, with vanishing integrals over balls of fixed radius and given majorant of growth.
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V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer, Dordrecht (2003).
V. V. Volchkov, “Final version of the local theorem on two radii,” Mat. Sb., 186, No. 6, 15–34 (1995).
I. D. Smith, “Harmonic analysis of scalar and vector fields in ℝn,” Proc. Cambridge Phil. Soc., 72, 403–416 (1972).
F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York (1955).
S. Thangavelu, “Spherical means and CR functions on the Heisenberg group,” J. Anal. Math., 63, 255–286 (1994).
V. V. Volchkov, “Solution of the problem of support for some classes of functions,” Mat. Sb., 188, No. 9, 13–30 (1997).
M. Shahshahani and A. Sitaram, “The Pompeiu problem in exterior domains in symmetric space,” Contemp. Math., 63, 267–277 (1987).
V. V. Volchkov, “Theorems on ball means on symmetric spaces,” Mat. Sb., 192, No. 9, 17–38 (2001).
O. A. Ochakovskaya, “On functions with vanishing integrals over balls of fixed radius on a half-plane,” Dokl. Ros. Akad. Nauk, 381, No. 6, 745–747 (2001).
O. A. Ochakovskaya, “On functions with vanishing integrals over balls of fixed radius,” Mat. Fiz. Anal. Geom., 9, No. 3, 493–501 (2002).
V. V. Volchkov, “On functions with vanishing integrals over cubes,” Ukr. Mat. Zh., 43, No. 6, 859–863 (1991).
V. V. Volchkov, “On the Pompeiu problem and its generalizations,” Ukr. Mat. Zh., 45, No. 10, 1444–1448 (1993).
V. V. Volchkov, “Spherical means on Euclidean spaces,” Ukr. Mat. Zh., 50, No. 10, 1310–1315 (1998).
V. E. Silenko, “New theorems of Morera type in a unit disk,” Ukr. Mat. Zh., 53, No. 2, 278–281 (2001).
N. P. Volchkova, “On the inversion of local Pompeiu transforms,” Ukr. Mat. Zh., 55, No. 7, 875–880 (2003).
B. G. Korenev, An Introduction to the Theory of Bessel Functions [in Russian], Nauka, Moscow (1971).
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Distribution Theory and Fourier Analysis, Springer, Berlin (1983).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 857–861, June, 2008.
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Ochakovskaya, O.A. Majorants of functions with vanishing integrals over balls. Ukr Math J 60, 1003–1008 (2008). https://doi.org/10.1007/s11253-008-0098-4
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DOI: https://doi.org/10.1007/s11253-008-0098-4