I have a function $A(x) = \int_{\partial B_1(0)} e^{ikx} B(k) dk$ in $3D$ and I want to find $B(k)$ for a given $A(x)$. How can I do that?

In my use case it would be enough to invert $R_n^m(x) = \int_{\partial B_1(0)} e^{ikx} S_n^m(k) dk$ with $R \in SO(3; \mathbb R) \forall x$. Any help for the special or general case would be appreciated.