I have a stationary time series $(X_t)$ with spectral density $f_X$, i.e. $$f_X(\lambda) = \frac{1}{2\pi}\sum_{h\in\mathbb{Z}} e^{-ih\lambda}\gamma_X(h)$$ where $\lambda \in (-\pi,\pi]$ and $\gamma_X(h)$ is the autocovariance of $X$ at lag $h$. Next I apply a linear filter to $X$ to obtain a new process denoted by $Y$ as follows $$Y_t = \sum_{j\in\mathbb{Z}}\psi_jX_{t-j}$$ The filter is absolutely summable, i.e. $\sum_{j\in\mathbb{Z}}\lvert\psi_j\rvert< \infty$. I also define an object named transfer function, denoted by $\psi(\lambda)$, for the filter as follows. $$\psi(\lambda) = \sum_{j\in\mathbb{Z}}\psi_je^{-ij\lambda}$$
What I want to do is to express the spectral density of $Y$ in terms of the transfer function of the filter and the spectral density of $X$.
There are a couple of things that I know about $Y$. First of all, $X$ being stationary and the filter being absolutely summable imply that $Y$ is stationary as well. Furthermore, the autocovariance function of $Y$ is given by $$\gamma_Y(h) = \sum_j\sum_k\psi_j\psi_k\gamma_X(h-j+k)$$
Then by definition of spectral density $$f_Y(\lambda) = \frac{1}{2\pi}\sum_{h} e^{-ih\lambda}\gamma_Y(h) = \frac{1}{2\pi}\sum_{h} e^{-ih\lambda}\sum_j\sum_k\psi_j\psi_k\gamma_X(h-j+k)$$
I need a change of variables here. So I let $m = h-j+k$. Then, $$f_Y(\lambda) =\frac{1}{2\pi}\sum_{m} \sum_j\sum_ke^{-i(m+j-k)\lambda}\psi_j\psi_k\gamma_X(m)$$ I will now separate this triple sum and I think I am making a mistake here. $$f_Y(\lambda) =\frac{1}{2\pi}\sum_{m} \sum_j\sum_ke^{-im\lambda}e^{-ij\lambda}e^{ik\lambda}\psi_j\psi_k\gamma_X(m)$$
$$f_Y(\lambda) =\underbrace{\frac{1}{2\pi}\sum_me^{-im\lambda}\gamma_X(m)}_{f_X(\lambda)} \underbrace{\sum_je^{-ij\lambda}\psi_j}_{\psi(\lambda)}\underbrace{\sum_ke^{ik\lambda}\psi_k}_{\overline{\psi(\lambda)}}$$
$$f_Y(\lambda) = f_X(\lambda)\lvert\psi(\lambda)\rvert^2$$
The final answer seems correct but I can't help but feel that I skipped an important argument in the derivation. Could someone please take a look at this and make it rigorous if needed?