= Solution
For a <weakly stationary process> with mean $m$, the <autocovariance function> is $\gamma_k=\mathbb E[(X_{t+k}-m)(X_t-m)]$, independent of $t$. The positive definiteness of these covariances gives a unique finite positive <spectral measure of a stationary time series> on the frequency circle. Its cumulative function is the <time-series spectral distribution> $F$, with total mass $\gamma_0$, and
$$
\boxed{\gamma_k=\int_{-\pi}^{\pi}e^{ik\lambda}\,dF(\lambda).}
$$
Use one representative of the identified endpoints $-\pi$ and $\pi$ when specifying atoms. For a real-valued process the measure is symmetric, so the imaginary part integrates to zero. If it has a <time-series spectral density>, then
$$
\boxed{\gamma_k=\int_{-\pi}^{\pi}e^{ik\lambda}f(\lambda)d\lambda,\qquad
f(\lambda)=\frac1{2\pi}\sum_{k\in\mathbb Z}\gamma_k e^{-ik\lambda}.}
$$
The second identity is pointwise for an absolutely summable <covariance> sequence, as will apply to part (a). Existence of a density alone does not guarantee ordinary pointwise convergence of its <Fourier series>; in general the reconstruction can be interpreted by its <Cesàro means> in $L^1$. This fixes the frequency normalization for all three calculations below.
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