Sometimes decomposition into complex Fourir series is called discrete Fourier transform. Namely, consider $L$-periodic function (assuming for simplicity that it is sufficiently smooth) $f(x)$. Then it can be decomposed into Fourier series, which, following Section 5.1 we write as
\begin{align}
f(x)=&\sum _{k_n} C(k_n) e^{k_n x}\Delta k_n,
\label{eq-5.2C.1}\\
C(k):=&\frac{1}{2\pi}\int_I e^{-ik_nx}\,dx
\label{eq-5.2C.2}
\end{align}
with $k_n= \frac{2\pi i n}{L}$, $h=\Delta k_k= \frac{2\pi}{L}$. Now $k$ is a discrete variable, running lattice (grid) $K:=\frac{2\pi}{L}\mathbb{Z}$.

Let us call $C(k)$ discrete Fourier transform.

Which properties of Fourier transform (see Section 5.2) hold for discrete Fourier transfor, and which should be modified?

From Theorem 5.2.3 survive Statements 1, 2 (with $b\in K$ now, so periodicity does not break), 3, 5 (but only with $\lambda\in \mathbb{Z}$, so periodicity does not break).

Statement 4 obviously does not, since multiplication by $x$ breaks periodicity. Instead consider multiplications by
$\frac{1}{hi} \bigl(e^{ixh}-1\bigr)$,
$\frac{1}{hi} \bigl(1-e^{-ixh}\bigr)$ and
$\frac{1}{h} \sin (xh)$. Due to Property 2 these multiplications become finite differences
\begin{align*}
\Lambda^-_h\phi (k)=& \frac{1}{h}\bigl(\phi(k)-\phi(k-h)\bigr),\\
\Lambda^+_h\phi (k)=&\frac{1}{h}\bigl(\phi(k+h)-\phi(k)\bigr) ,\\
\Lambda^-_h\phi (k)=& \frac{1}{2h}\bigl(\phi(k+h)-\phi(k-h)\bigr)
\end{align*}
while multiplication by $\frac{2}{h^2}(\cos (xh)-1)$ becomes a discrete Laplacian
\begin{equation*}
\Delta_h \phi (k) =
\frac{1}{h^2}\bigl(\phi(k+h)-2\phi(k)+\phi(k-h)\bigr).
\end{equation*}

From Theorem 5.2.4 Statement 1 does not survive since convolution of periodic functions is not necessarily defined, but Statement 2 survives with a discrete convolution
\begin{equation*}
(f*g)(k)=\sum _{\omega \in K} f(\omega)g(k-\omega)h.
\end{equation*}

Those who are familiar with distributions (see Section 11.1 can obseve that for $L$-periodic functions ordinary Fourier transform is
\begin{equation*}
\hat{f}(\omega ) = \sum_{k\in K} C(k) \delta (\omega -k) h
\end{equation*}
where $\delta$ is a Dirac delta-function.