Suppose that $X_t$ is a weakly stationary process; then, its autocovariance function can be represented as: $$\gamma_X(h) = \int_{(-\pi , \pi ]} e^{i h v} dF(v)$$ The function $F$ is called the spectral distribution of $X_t$. If $F$ admits a density with respect to the Lebesgue measure, say $f$, we say that $f$ is the spectral density of $X_t$. I am interested in the following problem (taken from Brockwell and Davis, Time Series: Theory and Methods, Exercise 4.15):
Suppose $x_t = w_t - 2w_{t-1}$, where $w_t$ is a mean zero white noise sequence with variance $\sigma_w^2$. Given $\epsilon > 0$, find an integer $k$ and constants $a_0, \ldots , a_k$, with $a_0=1$, such that if $f_y$ is the spectral density of the process \begin{align*} y_t = \sum_{j=0}^k a_j x_{t-j} \end{align*} then, \begin{align*} \sup_{-0.5 \leq \omega \leq 0.5} \left| f_y(\omega ) - \frac{\mathrm{Var}({y_t})}{2\pi} \right| < \epsilon \end{align*}
Motivation: The key to this problem should be that for every symmetric spectral density (e.g. the variance of $y_t$), there exists an invertible $MA(q)$ process $X_t$ such that the spectral density of $X_t$ is within an epsilon of the initial symmetric spectral density. However, the process in this question takes away some freedom from which MA processes I can choose from to approximate the variance of $y_t$, which makes it non-obvious how to calculate the desired coefficients.