(1) The span of the trigonometric polynomials is $span \{ e^{i n x}\ |\ n \in \mathbb{Z} \} = L^2([-\pi,\pi])$. Is there any nice way of characterizing the span with a cutoff, namely: $span\{ e^{i n x}\ |\ n \in \mathbb{Z} \text{ and } |n| < N \}$ for some fixed $N>0$ ?
(2) Analogous question for the Fourier transform: given a fixed $\Lambda >0$, is there a nice way to characterize the functions $f$ that can be written as $f(x) = \int_{-\Lambda}^{\Lambda} dw\ e^{i w x} g(w)$ for some $g \in L^2([-\Lambda,\Lambda])$?
My naive guess for (2) would be $\{ f \in L^2(\mathbb{R})\ |\ f \text{ is Lipshitz continuous with constant } \Lambda \}$. Intuitively, a cutoff in the frequency domain ($w$) should lead to a maximum resolution in the signal domain ($x$).
Feel free to answer for pointwise convergence, $L^2$ convergence, or both; assume $N$ and $\Lambda$ are ''large'' if necessary (if there is a way to make sense of that).
(1) It's just the trigonometric polynomials of specified length. These are finite dimensional subspaces of $L^2[-\pi \pi]$ in your case.
(2) It's the space of "band-limited" $L^2$-functions. Also a Hilbert subspace, they are characterized by Shannon sampling theorem and Paley-Wiener theorem. They are smooth, in fact extendable to an entire function of exponential type.