Approximation of functions on rays in $\Bbb{R}^n$ by solutions to convolution equations

This is a first study of approximation of continuous functions on rays in $\Bbb{R}^n$ by smooth solutions to a multidimensional convolution equation with a radial convolutor. We obtain an analog of the well-known Carleman's Theorem on tangent approximation by entire functions. As consequences, we give some new results of interest for the theory of convolution equations. These results concern the density in $\Bbb{C}$ of the range of some solutions to the convolution equation as well as the possible growth of solutions on rays in $\Bbb{R}^n$.