Context: I am currently working through Chapter $8$ of Anders Vretblad's Fourier Analysis and Its Applications. This particular chapter focuses on distributions, and builds up to the Fourier transform of such. I am currently looking at Chapter $8.7$, focused on periodic distributions.
Relevant Definitions:
The Fourier transform and its inverse (for ordinary functions $\mathbb{R} \to \mathbb{C}$) are defined in the following ways for this text, in one-dimension: $$ \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\vp}{\varphi} \newcommand{\SS}{\mathcal{S}} \newcommand{\CC}{\mathcal{C}} \newcommand{\FF}{\mathcal{F}} \newcommand{\dd}{\mathrm{d}} \newcommand{\f}{\widehat{f}} \newcommand{\w}{\widehat} \begin{align*} \FF[f](\omega) &:= \int_\R f(x) e^{-i \omega x} \, \dd x \\ \FF^{-1}[f](x) &:= \frac{1}{2\pi} \int_\R f(\omega) e^{i \omega x} \, \dd \omega \end{align*}$$ We may sometimes define $\f := \FF[f]$ in general.
A sufficiently-nice function $ f : \R \to \C$ is identified identically with its distribution $T_f : \CC_c^\infty(\R) \to \C$ given by $T_f[\vp] := \int_\R f \vp$, i.e. we may refer to $f$ both as the function $f$ and the distribution $T_f$.
Schwartz Space: The Schwartz space $\mathcal{S} := \mathcal{S}(\R)$ is defined as the set of all $\vp \in \CC^\infty(\R)$ such that, for any $n,k \in \Z_{\ge 0}$, we can find constants $C_{n,k}$ such that $$ \left| \vp^{(k)}(x) \right| \le \frac{C_{n,k}}{ \big( 1 + |x| \big)^n } $$ (i.e. $\SS$ consists of rapidly-decaying smooth functions).
Tempered Distributions: The topological dual of the Schwartz class, $\SS'$, is known as the tempered distributions. Hence $f \in \SS'$ if and only if it is linear as a function $f : \SS \to \C$, and continuous in the sense of sequential convergence. That is, $f : \SS \to \C$ is continuous if and only if $\,\forall\{\vp_n\}_{n \in \N} \subseteq \SS$ and $\vp \in \SS$ such that $$ \vp_j \xrightarrow[\SS]{j\to\infty} \vp; \;\;\; \text{ i.e., } \;\;\; \forall n,k \in \Z_{\ge 0}, \; \lim_{j \to \infty} \sup_{x \in \R} \big( 1 + |x| \big)^n \left| \vp_j^{(k)} - \vp^{(k)} \right| = 0 $$ then we have $f[\vp_n] \xrightarrow[\C]{n \to \infty} f[\vp]$.
Translation: As a shorthand, the text likes to - given a function $f$ and $a \in \R$ - define $f_a$ to be the function $f_a(x) := f(x-a)$.
Periodic Distributions: A tempered distribution $f$ is said to be $p$-periodic if $f[\vp_p] = f[\vp]$ for every $\vp \in \mathcal{S}$.
Simple Zeroes: A complex function $f : \C \to \C$ is said to have a simple zero $\zeta \in \C$ if $f(z)/(z - \zeta)$ is complex-differentiable and nonzero in an open ball about $\zeta$.
Fourier Transform of a Tempered Distribution: The Fourier transform $\f$ of $f \in \SS'$ is defined by satisfying a Plancherel-like formula: $\f$ is the distribution in $\SS'$ such that $$ \f[\vp] = f[\w{\vp}] \text{ for all } \vp \in \CC_c^\infty(\R) $$
Background & Questions:
We focus on $2\pi$-periodic $f \in \SS'$ for this section, and so shall I. The goal seems to be to establish a unifying relationship between Fourier series, and Fourier transforms of periodic distributions.
Starting at the relationship $$ f = f_{2\pi} \stackrel{\FF}{\implies} \f = \w{f_{2\pi}} = e^{-2 \pi i \omega} \f $$ where the final equality follows by a theorem (Theorem $8.3c$ of the text). From here, then, $$ (1 - e^{- 2 \pi i \omega}) \f = 0 $$ Hence, wherever $\f \not \equiv 0$, we have $\omega \in \Z$. Such $\omega$ are simple zeroes of the function $1 - e^{-2 \pi i \omega}$.
This is where the text begins to lose me.
Question $1$: The text claims that, consequently, $1 - e^{-2 \pi i \omega}$ "behaves like" $C_n(\omega - n)$ for $n \in \Z$ for some constant $C_n$. Why is this the case? I'm guessing that, since $n \in \Z$ forms a simple zero, then $$ \frac{1 - e^{- 2 \pi i \omega}}{\omega - n} = C_n \ne 0 $$ in an open region about $n$, and ... I'm not sure. As stated above, $C_n$ still depends on $\omega$. Perhaps we take a limit as $\omega \to n$ in $\C$ and make an appeal to a Laurent/Taylor series expansion in some respect?
Question $2$: Ignoring this, the textbook makes reference to another result, $x f(x) \equiv 0$ (as a distribution) if and only if $f(x) = C \cdot \delta(x)$, for some constant $C$ and $\delta$ the Dirac distribution. The proof is straightforward. The textbook tries to apply this in a "local sense" at each $\omega = n \in \Z$, and conclude $$ \f(\omega) = \sum_{n \in \Z} \gamma_n \delta_n(\omega) = \sum_{n \in \Z} \gamma_n \delta(\omega - n) $$ for to-be-determined constants $\gamma_n$, later found to match those from the exponential Fourier series expansion of $f$. However, I am lost as to how this is justified.
Any ideas as to how to justify these would be welcome.