Spectral density transformation under a linear filter
= Spectral density transformation under a linear filter
{title2=$f_Y(\omega)=|\alpha(\omega)|^2f_X(\omega)$}
Filtering multiplies the <time-series spectral density> by the squared modulus of the frequency response. Insert the spectral integral into the <covariance> double sum and interchange sums using absolute summability of the filter. This holds even when the input <covariances> are not absolutely summable, provided the spectral measure has a density.