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 $R \in \mathrm{SO}(3; \mathbb R)$, $\forall x$. Any help for the special or general case would be appreciated.
It is helpful to rewrite the integral over the surface of the unit sphere into an integral over the whole 3D space, using the delta function $\delta(k-1)$, $$A(x,y,z)=\int_{-\infty}^\infty dk_x\int_{-\infty}^\infty dk_y\int_{-\infty}^\infty dk_z \,e^{ik_x x+ik_y y+ik_z z}B(k_x,k_y,k_z)\delta\left(\sqrt{k_x^2+k_y^2+k_z^2}-1\right).$$ Inversion of the Fourier transform gives $$B(k_x,k_y,k_z)\delta\left(\sqrt{k_x^2+k_y^2+k_z^2}-1\right)=(2\pi)^{-3}\int_{-\infty}^\infty dx\int_{-\infty}^\infty dy\int_{-\infty}^\infty dz\,e^{-ik_x x-ik_y y-ik_z z}A(x,y,z).$$ Integration over the radial coordinate then produces the function $B$ itself, as a function of the angular coordinates.
