# Broadening spectral data by using FFT's

I obtain numerical discrete data of the form

$$S_{raw}(\omega) = \sum_{j}w_{j} \delta(\omega-\omega_{j})$$

to compare the result with experimental data the delta peaks need to be broadened according to

$$S_{broad}(\omega) = \int d\omega^{\prime} K(\omega,\omega^{\prime}) S_{raw}(\omega^{\prime})$$

where $$K(\omega, \omega^{\prime})$$ is the broadening kernel i.e.

$$K(\omega,\omega^{\prime}) = \frac{\theta(\omega\omega^{\prime})}{\sqrt{\pi}b|\omega|}\exp\left[-\left(\frac{1}{b}\ln\left|\frac{\omega}{\omega^{\prime}}\right| - \frac{b}{4}\right)^2\right].$$

If $$K(\omega,\omega^{\prime}) = K(\omega-\omega^{\prime})$$ the integral equation above can be interpreted as a convolution and can be efficiently calculated by means of the convolution theorem and FFTs. But the example kernel $K$ does not have this property.

I ask me now if there is still a way to compute the broadened function by exploiting the convolution theorem and fast FFT algorithms ?