This is a cross-post from [signal processing forum](https://dsp.stackexchange.com/questions/52272/maximum-magnitude-deviation-between-dft-and-dtft) 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](https://dsp.stackexchange.com/questions/17814/why-does-the-dft-assume-the-transformed-signal-is-periodic) and [here](https://dsp.stackexchange.com/questions/37967/basics-of-leakage-phenomena-in-dft-and-its-approximation-with-sinc-function/37971#37971). 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 $\gamma m\_{\textrm{d}} \geq m\_{\textrm{c}}$?*



Maximum Magnitude Deviation between DFT and DTFT
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:


1. $F(z)$ has complex coefficients. What is the bound for real coefficients?
2. How does this bound change if we add zero padding? It will tend to 1, but how quickly?
3. $F(z)$ gives the worst case bound, but how is this bound for random (Gaussian) distributed coefficients?


