  In this article, we develop comprehensive frequency domain methods for
estimating and inferring the second-order structure of spatial point processes.
The main element here is on utilizing the discrete Fourier transform (DFT) of
the point pattern and its tapered counterpart. Under second-order stationarity,
we show that both the DFTs and the tapered DFTs are asymptotically jointly
independent Gaussian even when the DFTs share the same limiting frequencies.
Based on these results, we establish an $\alpha$-mixing central limit theorem
for a statistic formulated as a quadratic form of the tapered DFT. As
applications, we derive the asymptotic distribution of the kernel spectral
density estimator and establish a frequency domain inferential method for
parametric stationary point processes. For the latter, the resulting model
parameter estimator is computationally tractable and yields meaningful
interpretations even in the case of model misspecification. We investigate the
finite sample performance of our estimator through simulations, considering
scenarios of both correctly specified and misspecified models. Furthermore, we
extend our proposed DFT-based frequency domain methods to a class of
non-stationary spatial point processes.
