Smoothed exponential sums: bounds and sources? Let $f:\mathbb{R}\to\mathbb{C}$ be differentiable $k$ times, with $f, f',\dotsc,f^{(k)}\in L^1$. Let $\alpha\in \mathbb{R}/\mathbb{Z}$, $\alpha\ne 0$. In "Every odd number..." (Math. Comp. 83, 2014), Lemma 3.1, Tao shows that
$$\left|\sum_{n\in \mathbb{Z}} f(n) e(\alpha n)\right|\leq \frac{1}{|2 \sin(\pi \alpha)|^k} |f^{(k)}|_1,$$
where $e(t) = e^{2\pi i t}$. 
The proof goes essentially by summation by parts.
(a) Are there older sources for this? Somewhat confusingly, Tao credits Gallagher ("The large sieve") and Lemma 1.1 in Montgomery's Topics in Multiplicative Number Theory, but they give only equation (3.1) in Tao's papers, not the inequality above.
(b) For $k=2$, this is not in general optimal: one can show 
$$\left|\sum_{n\in \mathbb{Z}} f(n) e(\alpha n)\right|\leq \frac{1}{|2 \sin(\pi \alpha)|^2} |\widehat{f''}|_\infty,$$
which is no weaker and often strictly stronger, since $|\widehat{f''}|_\infty\leq |f''|_1$.
This is Lemma 2.1 in my three-prime book draft on the arxiv; the proof I give goes by the Poisson summation formula, plus Euler's formula for the cotangent.
Are similar bounds true for general $k$? (Is $\left|\sum_{n\in \mathbb{Z}} f(n) e(\alpha n)\right| \leq |\widehat{f'}|_\infty/|2 \sin \pi \alpha|$, for instance?) Again, can such results be found in older sources?
 A: By Poisson:
$$ \sum_{n \in \mathbb Z} f(n) e(\alpha n)= \sum_{m \in \mathbb Z} \hat{f} ( 2 \pi m + \alpha) $$
By the formula for the derivative of the Fourier transform:
$$= \sum_{m \in \mathbb Z} \frac{1}{ (2\pi i m + \alpha i)^k}\widehat{f^{(k)}} ( 2 \pi m + \alpha) \leq \sum_{m \in \mathbb Z} \frac{1} {\left| 2 \pi m + \alpha \right|^k}\left|\widehat{f^{(k)}}\right|_{\infty}$$
if $k >1$ then this sum converges and we may replace your constant by $\sum_{m \in \mathbb Z} \frac{1} {\left| 2 \pi m + \alpha \right|^k}$. I'm not sure if there's an analytic formula for this for various values of $k$, we can certainly estimate.
If $k=1$ then this sum diverges and this method fails to prove a bound. Does this mean that no bound is posssible? Yes. Simply take a Schwartz function $g$ that does something nice and smooth on $[-1,1]$, behaves like $1/|\xi|$ on $[1,n]$ and $[-n,-1]$, and then declines rapidly to $0$ outside $[-n,n]$.  Then let $f$ be the inverse Fourier transform, also a Schwartz function, so $\hat{f}=g$. Then the Poisson summation formula is justified but:
$$\sum_{m \in \mathbb Z} \hat{f} ( 2 \pi m + \alpha)  \approx \log n$$
$$|\widehat{ f'} |_\infty \approx 1$$
So no bound in terms of $|\widehat{ f'} |_\infty$ is possible.
