We study generalizations of the class of functions with zero integrals over the balls of fixed radius. An analog of the John uniqueness theorem is obtained for weighted spherical means on a sphere.
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F. John, “Abhängigkeiten zwischen den Flächenintegralen einer stetigen Funktion,” Math. Ann., 111, No. 1, 541–559 (1935).
F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York (1955).
V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer, Dordrecht (2003).
J. D. Smith, “Harmonic analysis of scalar and vector fields in \( {{\mathbb{R}}^n} \),” Proc. Cambridge Phil. Soc., 72, 403–416 (1972).
B. Ya. Levin, Distribution of Roots of Entire Functions [in Russian], Gostekhizdat, Moscow (1956).
A. F. Leont’ev, Sequences of Polynomials of Exponents [in Russian], Nauka, Moscow (1980).
Yu. I. Lyubich, “On the uniqueness theorem for functions periodic in the mean,“ Zap. Nauch. Sem. LOMI, 81, 166 (1978).
P. P. Kargaev, “On the roots of functions periodic in the mean,” Mat. Zametki, 37, No. 3, 322–325 (1985).
P. Koosis, Introduction to H p Spaces With an Appendix on Wolff’s Proof of the Corona Theorem, Cambridge Univ. Press, Cambridge (1980).
E. T. Quinto, “Pompeiu transforms on geodesic spheres in real analytic manifolds,” Isr. J. Math., 84, 353–363 (1993).
D. A. Zaraiskii, “Improvement of the uniqueness theorem for the solutions of convolution equations,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 12 (2006) pp. 69–75.
V. V. Volchkov, “Uniqueness theorems for the solutions of convolution equations on symmetric spaces,” Izv. Ros. Akad. Nauk, Ser. Mat., 70, No. 6, 3–18 (2006).
V. V. Volchkov, “Local theorem on two radii on symmetric spaces,” Mat. Sb., 198, No. 11, 21–46 (2007).
Vit. V. Volchkov, “On functions with zero ball means on compact two-point homogeneous spaces,” Mat. Sb., 198, No. 4, 21–46 (2007).
V. V. Volchkov and Vit. V. Volchkov, Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group, Springer, London (2009).
L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. 1. Distribution Theory and Fourier Analysis, Springer, Berlin (1983).
V. V. Volchkov, “Theorem on mean for one class of polynomials,” Sib. Mat. Zh., 35, No. 4, 737–745 (1994).
P. Ungar, “Freak theorem about functions on a sphere,” J. London Math. Soc., 29, No. 1, 100–103 (1954).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1954).
N. Ya. Vilenkin, Special Functions and the Theory of Representations of Groups [in Russian], Nauka, Moscow (1991).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Additional Chapters [in Russian], Nauka, Moscow (1986).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 5, pp. 611–619, May, 2013.
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Volchkov, V.V., Savost’yanova, I.M. Analog of the john theorem for weighted spherical means on a sphere. Ukr Math J 65, 674–683 (2013). https://doi.org/10.1007/s11253-013-0805-7
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DOI: https://doi.org/10.1007/s11253-013-0805-7