5.2.C. Discrete Fourier transform

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

### 5.2.C. Discrete Fourier transform

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.