### An extension of deLeeuw’s theorem to the $n$-dimensional rotation group

We study a method of approximating representations of the group $M\left(n\right)$ by those of the group $SO(n+1)$. As a consequence we establish a version of a theorem of DeLeeuw for Fourier multipliers of ${L}^{p}$ that applies to the “restrictions” of a function on the dual of $M\left(n\right)$ to the dual of $SO(n+1)$.