Suppose a function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that:
- $|g(x)|\leq g(0)$;
- $g(x)=g(-x)$, i.e. $g(x)$ is even;
- $\int_{-\infty}^{\infty}g(x)dx=C$;
- There exists a Fourier transform of $g(x)$, $\mathcal{F}(g(x))=G(\xi)$. Since $g(x)$ is even, $G(\xi)$ is real, even, and non-negative. Also, the aforementioned constant $C=G(0)$.
Here, my $g(x)$ is an autocorrelation function and $G(\xi)$ is the corresponding power-spectral density.
Given the conditions above, is it possible that:
$\int_{-\infty}^{\infty}|g(x)|dx=\infty$
I can't think of an example $g(x)$ where this happens, nor of a proof that such $g(x)$ does not exist. Any help?