Let $f\in C^{\infty}_c(\mathbb R^2)$ be of the form $$f(z)=f_1(|z|)e^{in\theta}, \quad z=|z|e^{i\theta}.$$ In a paper, I found that using the Hecke-Bochner identity, we get $$\hat{f}(z)=\hat{f}(\lambda e^{i s})=\lambda^{|m|}g(\lambda)e^{ims}, $$ where $g(\lambda)$ is a constant times the $2 + 2|m|$−dimensional Fourier transform of the function $\frac{f_1(|z|)}{|z|^{|m|}}$ considered as a radial function on $\mathbb R^{2 + 2|m|}$.
What is the statement of this theorem (Hecke-Bochner identity) and what is this theorem for?