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On functions with zero integrals over ellipsoids. (English. Russian original) Zbl 1055.31004
Dokl. Math. 63, No. 1, 25-27 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 376, No. 2, 158-160 (2001).
A compact set $$A\subseteq\mathbb{R}^n$$ is said to be a Pompeiu set on the ball $$B_r=\{x\in\mathbb{R}^n:| x|<r\}$$ if every locally summable function $$f:B_r\to\mathbb{R}$$ such that $$\int_{\lambda A}f(x)\,dx=0$$ for all motions $$\lambda$$ of $$\mathbb{R}^n$$ for which $$\lambda A\subset B_r$$ vanishes almost everywhere. For many compact sets $$A$$, this holds if $$r$$ is sufficiently large. The author stated the following problem in [Sb. Math. 189, No. 7, 955–976 (1997; Zbl 0957.53042)]: For a given compact set $$A$$, determine $$r(A)=\inf\{r>0:A\in{\mathcal P}(B_r)\}$$, where $${\mathcal P} (B_r)$$ is the totality of all Pompeiu sets on $$B_r$$. Some upper estimates for $$r(A)$$ have been found by C. A. Berenstein, R. Gay and A. Yger [J. Anal. Math. 54, 259–287 (1990; Zbl 0723.44002)]. Here the author determines $$r(A)$$ for all ellipsoids $$A$$ that are not balls. He then applies his techniques to solving other problems concerning functions with vanishing integrals over ellipsoids.
##### MSC:
 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 26B15 Integration of real functions of several variables: length, area, volume 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
##### Keywords:
Pompeiu problem; ellipsoid; inverse problem; Pompeiu set