in several time series texts like Shumway, the Fourier transform of the autocovariance function is integrated only over frequencies from $-\frac{1}{2}$ to $\frac{1}{2}$. I do not understand why it doesn't integrate over $[-\infty, \infty]$ but I vaguely remember something from my numerical methods course about the interval and periodicity. (Could it be that this assumes a period of $1$?)
Could someone explain why we don't have to consider $f$ outside of $[-\frac{1}{2}, \frac{1}{2}]$ here:
$$ \gamma(h) = \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{i2 \pi hf}df$$ where $f$ is the frequency and $h$ the lag?
many thanks in advance!