= Solution
The series defining the <linear filter of a stationary time series> converges in $L^2$, because the norm of a tail is at most $\|X_0\|_2$ times the corresponding tail sum of $|a_s|$. Its mean is the original mean times $\sum_s a_s$. Its <covariance> is time independent and equals
$$
\gamma_k^Y=\sum_{s,u\in\mathbb Z}a_sa_u\gamma_{k-s+u}.
$$
Under the summability assumption in (i),
$$
\sum_k|\gamma_k^Y|
\leq\left(\sum_s|a_s|\right)^2\sum_k|\gamma_k|<\infty.
$$
We may therefore interchange all the sums. Setting $h=k-s+u$ in the Fourier series gives
$$
\begin{aligned}
f_Y(\omega)
&=\frac1\pi\sum_{s,u}a_sa_u e^{i(s-u)\omega}
\sum_h\gamma_he^{ih\omega}\\
&=\left(\sum_sa_se^{is\omega}\right)
\left(\sum_ua_ue^{-iu\omega}\right)f_X(\omega).
\end{aligned}
$$
The second factor is the conjugate of the first because the coefficients are real. Hence
$$
\boxed{f_Y(\omega)=|\alpha(\omega)|^2f_X(\omega).}
$$
This <spectral density transformation under a linear filter> also holds whenever the original spectral measure has a density, without <covariance> summability: insert its integral representation into the <covariance> double sum, justified by absolute summability of the filter coefficients.
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