Moment conditions for functions with zero integrals over congruent balls

Author V. V. Volchkov, Vit. V. Volchkov
valeriyvolchkov@gmail.com
Department of Mathematics and Information Technologies, Donetsk National University

Abstract We consider the question of precise conditions ensuring that a function having zero integrals over all balls of fixed radius is equal to zero. We completely investigate the case where together with zero integrals over congruent balls a function has zero first moments over these balls.
Keywords spherical means; mean periodicity
Reference 1. Volchkov V.V. Integral Geometry and Convolution Equations. Dordrecht: Kluwer Acad. Publ., 2003. 454 p.

2. Volchkov V.V., Volchkov Vit.V. Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group. London: Springer, 2009. 671 p.

3. Zalcman L. A bibliographic survey of the Pompeiu problem in: Fuglede B. et al. (eds.), Approximation by Solutions of Partial Differential Equations. Dordrecht: Kluwer Acad. Publ., 1992. P. 185194.

4. Zalcman L. Supplementary bibliography to a bibliographic survey of the Pompeiu problem// Radon Transforms and Tomography, Contemp. Math. 2001. V.278. P. 6974.

5. Stein E., Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. New Jersey: Princeton University Press, 1971. 332 p.

6. Vilenkin N.Y. Special Functions and the Theory of Group Representations. Moscow: Nauka, 1991. 576 p.

7. Treves J.F. Lectures on Linear Partial Differential Equations with Constant Coefficients. Rio de Janeiro: Fasiculo Publicado Pelo Instituto de Matematica Pura e Aplicada do Conselho Nacional de Pesquisas, 1961. 243 p.

8. Suetin P.K. Classical Orthogonal Polynomials. Moscow: Nauka, 1979. 328 p. (in Russian)

9. Volchkov V.V., Volchkov Vit.V. Convolution equations on multidimensional domains and the reduced Heisenberg group// Mat. Sbornik. 2008. V.199, 8. P. 2960.

Pages 84-92
Volume 39
Issue 1
Year 2013
Journal Matematychni Studii
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