Symmetry problems in harmonic analysis

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 boundary $S$, $j_0(r)$ be the spherical Bessel function. The harmonic analysis symmetry problems are stated in the following theorems: {\bf Theorem A.} {\em 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 $j_0(ka)=0$. } {\bf Theorem B.} {\em Assume that ${\int }_D{e}^{ik\beta \cdot x}dx=0$ for all $\beta \in S^2$. Then $D$ is a ball.