Spherical means in annular regions. (English) Zbl 0841.31006

Let \(A(r, R)\) denote the annular region \(\{x\in \mathbb{R}^n: r<|x|< R\}\), where \(0\leq r< R\leq\infty\). A function \(f\) of class \(C^0 (A(r, R))\) is said to belong to \(Z(A (r, R))\) if \(\int_{\partial B} f d\sigma=0\) for every ball \(B\) such that \(\partial B\subset A(r, R)\) and \(0\in B\); here \(\sigma\) denotes surface measure. The vector space \(Z(A (r, R))\) is characterized in terms of projections into subspaces consisting of spherical harmonics. In particular, it is shown that \(Z(A (r,R))\) is infinite-dimensional and that \(Z(A (r,\infty))\) contains infinite-dimensional subspaces of functions with polynomial rates of decay. The latter result contrasts with known uniqueness results for the Radon transform of rapidly decaying functions of class \(C^0 (A (r, \infty))\).


31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
33C55 Spherical harmonics
44A99 Integral transforms, operational calculus
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