Let $\ell^1(\mathbb{Z})$ be the space of biinfinite sequences $f = (f(n))_{n \in \mathbb{Z}} \subset \mathbb{C}$ such that it is absolutely summable. The discrete Fourier transform or Fourier series is defined by, $$ \mathcal{F}(f)(e^{i \theta}) = \sum_{n \in \mathbb{Z}} f(n) e^{-in\theta}, \quad \theta \in (-\pi,\pi]. $$ The inverse discrete Fourier transform is defined as, $$ \mathcal{F}^{-1}(F)(n) = \frac{1}{2\pi} \int_{-\pi}^\pi F(e^{i \theta}) e^{in\theta} \, d\theta, \quad n \in \mathbb{Z}. $$
On the other hand, in asymptotics theory, we have the stationary phase approximation, where integrals of the form, $$ \int_\mathbb{R} g(x) e^{ikf(x)} \, dx, $$ can be estimated when $k \to \infty$, provided all critical points of $f$ are nondegenerate, i.e., $f''(x) \neq 0$. Hence, this theory doesn't directly apply to the inverse discrete Fourier transform above.
Question
Is there a theory that applies to the inverse discrete Fourier transform when $n \to \infty$ for asymptotic results? Any book or article recommendations on this topic?
