The dual horospherical Radon transform for polynomials

Let $X=G/K$ be a semisimple symmetric space of non-compact type. A horosphere in $X$ is an orbit of a maximal unipotent subgroup of $G$. The set $\operatorname{Hor}X$ of all horospheres is a homogeneous space of $G$. The horospherical Radom transform suggested by I. M. Gelfand and M. I. Graev in 1959 takes any function $\varphi$ on $X$ to a function on $\operatorname{Hor}X$ obtained by integrating $\varphi$ over horospheres. We explicitly describe the dual transform in terms of its action on polynomial functions on $\operatorname{Hor}X$.