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?