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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
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