Abstract
Symmetry problems in harmonic analysis are formulated and solved. One of these problems is equivalent to the refined Schiffer’s conjecture which was recently proved by the author. Let \(k=const>0\) be fixed, \(S^2\) be the unit sphere in \({\mathbb R}^3\), D be a connected bounded domain with \(C^2-\)smooth connected boundary S, \(j_0(r)\) be the spherical Bessel function. The harmonic analysis symmetry problems are stated in the following theorems. Theorem A Assume that \(\int _S e^{ik\beta \cdot s}ds=0\) for all \(\beta \in S^2\). Then S is a sphere of radius a, where \(j_0(ka)=0\). Theorem B Assume that \(\int _D e^{ik\beta \cdot x}dx=0\) for all \(\beta \in S^2\). Then D is a ball.
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Ramm, A.G. Symmetry problems in harmonic analysis. SeMA 78, 155–158 (2021). https://doi.org/10.1007/s40324-020-00235-w
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DOI: https://doi.org/10.1007/s40324-020-00235-w