A uniqueness theorem for mean periodic functions on the Bessel – Kingmann hypergroup

One of the properties of a periodic function on the real axis is that it is completely determined by its values on the period. This fact admits the following nontrivial multidimensional generalization: if a function $f\in C^\infty (\mathbb R^n)$ $(n\ge 2)$ with zero integrals over all spheres (or balls) of fixed radius $r$ is zero in some ball of radius $r$, then $f$ is zero in $\mathbb R^n$. The condition of infinite smoothness of the function $f$ in this statement cannot be relaxed. In this paper, we study a similar phenomenon for solutions of convolution equations related to the generalized Bessel shift operator. First, we consider the case when the convolution factor in the equation is an indicator of a segment symmetric with respect to zero. It is shown that the solutions to such an equation are determined by their values on the specified segment. Further, a generalization of  this property for the general Bessel convolution equation is given. The results obtained are analogues of the well-known uniqueness theorems for mean periodic functions belonging to F. John, Yu.I. Lyubich and A.F. Leontiev.