Maximum Magnitude Deviation between DFT and DTFT This is a cross-post from signal processing forum as it was not conclusive. 
Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then
$$
X(\omega) = \sum_{n = 0}^{N-1} x[n] e^{-j \omega n}.
$$
The length-$N$ DFT of $x[n]$ is
$$
X[k] = \sum_{n = 0}^{N-1} x[n] e^{-j 2 \pi \frac{k n}{N}}.
$$
For this, the DFT is a sampled version of the DTFT, i.e., 
$$
X[k] = X(2\pi k / N).
$$
Please also see posts here and here. Now, we consider the maximum magnitudes $m_{\textrm{d}} = \max_k |X[k]|$ and $m_{\textrm{c}} = \max_\omega |X(\omega)|$. Because of above, we have $m_{\textrm{d}} \leq m_{\textrm{c}}$.
How much does $m_{\textrm{d}}$  underestimate $m_{\textrm{c}}$? Is there $\gamma > 1$ such that $\gamma m_{\textrm{d}} \geq m_{\textrm{c}}$?
 A: I believe to have found a partial answer in the paper:
Gronwall, T. H. (1921). A sequence of polynomials connected with the $n$th roots of unity. Bulletin of the American Mathematical Society, 27(6), 275–279. http://doi.org/10.1090/S0002-9904-1921-03411-2
It actually gives a full construction of a polynomial $F(z)$ such that $|F(e^{i2\pi k/N})| \leq 1$, but $\max |F(z)| = \frac{1}{N} \sum_{n=0}^{N-1} \frac{1}{\sin \frac{2n + 1}{2N}\pi}$.
This makes the upper bound $\gamma$ in the original question to be asymptotically equal to 
$$
\frac{2}{\pi}\left( \log N + C + \log \frac{2}{\pi} \right) + o(1),
$$
where $C$ is the Euler's constant, and $o(1)$ tends to zero as $N$ increases indefinitely. There are various follow up questions:


*

*$F(z)$ has complex coefficients. What is the bound for real coefficients?

*How does this bound change if we add zero padding? It will tend to 1, but how quickly?

*$F(z)$ gives the worst case bound, but how is this bound for random (Gaussian) distributed coefficients?

