$f(\stackrel{\u20d7}{r})=f(r,{\psi}_{r},{\theta}_{r})=\sum _{L}{f}_{l}^{k}(r){Y}_{l}^{k}({\psi}_{r},{\theta}_{r})$

${\int}_{0}^{2\pi}{\int}_{0}^{\pi}f(\stackrel{\u20d7}{r})\overline{{Y}_{l}^{k}({\psi}_{r},{\theta}_{r})}\text{d}{\Omega}_{r}$

$4\pi {(i)}^{l}{\int}_{0}^{\infty}{f}_{l}^{k}(r){j}_{l}(\rho r){r}^{2}\text{d}r$

$F(\stackrel{\u20d7}{\omega})=F(\rho ,{\psi}_{\omega},{\theta}_{\omega})=\sum _{L}{F}_{l}^{k}(\rho ){Y}_{l}^{k}({\psi}_{\omega},{\theta}_{\omega})$

   
$\delta (\stackrel{\u20d7}{r}{\stackrel{\u20d7}{r}}_{0})=\delta (r{r}_{0})/({r}^{2}\text{\hspace{0.17em} sin}{\psi}_{r}).\delta ({\psi}_{r}{\psi}_{0})\delta ({\theta}_{r}{\theta}_{0})$

$\delta (r{r}_{0})/{r}^{2}\overline{{Y}_{l}^{k}({\psi}_{{r}_{0}},{\theta}_{{r}_{0}})}$

$4\pi {(i)}^{l}{j}_{l}(\rho {r}_{0})\overline{{Y}_{l}^{k}({\psi}_{{r}_{0}},{\theta}_{{r}_{0}})}$

${e}^{i\stackrel{\u20d7}{\omega}\cdot {\stackrel{\u20d7}{r}}_{0}}$

   
${e}^{i{\stackrel{\u20d7}{\omega}}_{0}\cdot \stackrel{\u20d7}{r}}$

$4\pi {(i)}^{l}{j}_{l}({\rho}_{0}r)\overline{{Y}_{l}^{k}({\psi}_{{\omega}_{0}},{\theta}_{{\omega}_{0}})}$

${(2\pi )/}^{3}{\rho}^{2}\delta (\rho {\rho}_{0})\overline{{Y}_{l}^{k}({\psi}_{{\omega}_{0}},{\theta}_{{\omega}_{0}})}$

${(2\pi )}^{3}\delta (\stackrel{\u20d7}{\omega}{\stackrel{\u20d7}{\omega}}_{0})$

   
$f(\stackrel{\u20d7}{r}{\stackrel{\u20d7}{r}}_{0})$

$\sum _{{l}^{\prime}=0}^{\infty}\phantom{\rule{0.2em}{0ex}}\sum _{{k}^{\prime}={l}^{\prime}}^{{l}^{\prime}}8{(i)}^{l{l}^{\prime}}\overline{{Y}_{{l}^{\prime}}^{{k}^{\prime}}({\psi}_{{r}_{0}},{\theta}_{{r}_{0}})}\phantom{\rule{0.1em}{0ex}}\sum _{{l}^{\u2033}=l{l}^{\prime}}^{l+{l}^{\prime}}{(i)}^{{l}^{\u2033}}\cdot {c}^{{l}^{\u2033}}(l,k,{l}^{\prime},{k}^{\prime}){\int}_{0}^{\infty}{f}_{{l}^{\u2033}}^{k{k}^{\prime}}(u){S}_{{l}^{\u2033}}^{l,{l}^{\prime}}(u,r,{r}_{0}){u}^{2}\text{d}u$

$[4\pi {(i)}^{l}{j}_{l}(\rho {r}_{0})\overline{{Y}_{l}^{k}({\psi}_{{r}_{0}},{\theta}_{{r}_{0}})}]\ast {F}_{l}^{k}(\rho )$

${e}^{i\stackrel{\u20d7}{\omega}\cdot {\stackrel{\u20d7}{r}}_{0}}F(\stackrel{\u20d7}{\omega})$

   
$f(r)g(r)$

${h}_{l}^{k}(r)={f}_{l}^{k}(r)\ast {g}_{l}^{k}(r)\u2254\sum _{{L}^{\prime}}{f}_{{l}^{\prime}}^{{k}^{\prime}}\sum _{{l}^{\u2033}=l{l}^{\prime}}^{l+{l}^{\prime}}{c}^{{l}^{\u2033}}(l,k,{l}^{\prime},{k}^{\prime}){g}_{{l}^{\u2033}}^{k{k}^{\prime}}$

$4\pi {(i)}^{l}{\int}_{0}^{\infty}{h}_{l}^{k}(r){j}_{l}(\rho r){r}^{2}\text{d}r$

$G(\stackrel{\u20d7}{\omega})\ast \ast \ast F(\stackrel{\u20d7}{\omega})$

   
$f(r)\ast \ast \ast g(r)$

${i}^{l}/2{\pi}^{2}{\int}_{0}^{\infty}{H}_{l}^{k}(\rho ){j}_{l}(\rho r){\rho}^{2}\text{d}\rho $

${H}_{l}^{k}(\rho )={G}_{l}^{k}(\rho )\ast {F}_{l}^{k}(\rho )$

$G(\stackrel{\u20d7}{\omega})F(\stackrel{\u20d7}{\omega})$

   
$f(\stackrel{\u20d7}{r}){\ast}_{(\psi ,\theta )}g(\stackrel{\u20d7}{r})$

${h}_{l}^{k}(r)={f}_{l}^{k}(r){g}_{l}^{k}(r)$

$4\pi {(i)}^{l}{\int}_{0}^{\infty}{h}_{l}^{k}(r){j}_{l}(\rho r){r}^{2}\text{d}r$

$\sum _{L}{H}_{l}^{k}{(\rho )Y}_{l}^{k}({\psi}_{\omega},{\theta}_{\omega})$

   
$(f{\ast}_{R}g)(r,{\psi}_{r},{\theta}_{r})$

${h}_{l}^{k}(r)={f}_{l}^{k}(r){g}_{l}^{0}(r){(1)}^{k}\sqrt{4\pi /2l+1}$

$4\pi {(i)}^{l}{\int}_{0}^{\infty}{h}_{l}^{k}(r){j}_{l}(\rho r){r}^{2}\text{d}r$

$\sum _{L}{H}_{l}^{k}(\rho ){Y}_{l}^{k}({\psi}_{\omega},{\theta}_{\omega})$

   