In spectral density theorem we often found spectral density in a form of series with delta function
$$J(\omega)=\sum_{k=0}^\infty \frac{c_k^2}{m_k \omega_k}\delta(\omega-\omega_k)$$
If I want to define a "squared" of spectral density as
$$J^2(\omega)=\sum_{k=0}^\infty \frac{c_k^4}{m_k^2 \omega_k^2}\delta(\omega-\omega_k)$$
How we define a relationship between the two ?
Attempt at solution :
If I define $J^2(\omega)$ as,
$$J^2(\omega)=J(\omega) \times J(\omega) \times \delta_{kk'}$$ $$J^2(\omega)=(\sum_{k=0}^\infty \frac{c_k^2}{m_k \omega_k}\delta(\omega-\omega_k))(\sum_{k'=0}^\infty \frac{c_{k'}^2}{m_{k'} \omega_{k'}}\delta(\omega-\omega_{k'}))\delta_{kk'}$$ $$J^2(\omega)=\sum_{k=0}^\infty \frac{c_k^4}{m_k^2 \omega_k^2}\delta(\omega-\omega_k)\delta(\omega-\omega_k)$$
It contains squared of delta function $\delta^2(\omega-\omega_k)$ which is undefined.
How then we can squared a series that contain delta function as in this example case, the series we usually define as spectral density ?