Abstract
Functions are investigated whose integrals over a given collection of sets are zero. Pompeiu sets are described in terms of the approximation of their indicators by linear combinations of the indicators of balls with special radii.
References
V. P. Zastavnyi and R. M. Trigub, “On functions with zero integrals over the sets congruent to a given set,” in:Theory of Functions and Approximations, Proc. of the Third Saratov Winter School, [in Russian], Part 2, Saratov University, Saratov (1986) p. 240.
S. Williams, “A partial solution of the Pompeiu problem,”Math. Ann.,223, 84–91 (1976).
P. G. Laird, “A reconsideration of ‘three squares’ problem,”Aequat. Mat.,21, No. 1, 98–104 (1980).
V. P. Zastavnyi,A Theorem on the Zeros of the Fourier Transform of an Indicator and Its Applications [in Russian], Deposited in VINITI No. 701-V.87, Novosibirsk (1986).
V. V. Volchkov,On Functions with Zero Integrals over Certain Sets [in Russian], Deposited at UkrNIINTI No. 301-Uk91, Donetsk (1991).
S. Helgason,Groups and Geometric Analysis, Academic Press, New York (1984).
B. G. Korenev,An Introduction to the Theory of Bessel Functions [in Russian], Nauka, Moscow (1971).
V. V. Volchkov, “On functions with zero integrals over cubes,”Ukr. Mat. Zh.,43, No. 6, 859–863 (1991).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1444–1448, October, 1993.
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Volchkov, V.V. On the Pompeiu problem and its generalizations. Ukr Math J 45, 1623–1628 (1993). https://doi.org/10.1007/BF01571095
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DOI: https://doi.org/10.1007/BF01571095