## 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 $$C^0 (A (r, \infty))$$.

### MSC:

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

### Keywords:

mean value; spherical harmonics; Radon transform
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### References:

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