Interpolation using trigonometric polynomials of bounded modulus

Question

Consider a grid of points $T=\{t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to derive conditions on $t_1,\ldots,t_m$ (interpolation points) under which for any sequence of complex numbers $c_1,c_2,\ldots,c_m\in\mathbb{C}$ with $|c_k|=1$, there exists a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form \begin{equation*} f(t)=\sum_{k=-n}^n a_k e^{2\pi i k t} \end{equation*}

(with n as small as possible) such that

1) $f(t)$ interpolates $c_1,c_2,\ldots,c_m\in\mathbb{C}$ on $T$. That is \begin{equation*} f(t_k)=c_k \end{equation*}

2) $|f(t)|< 1$ for $t\notin T$.

I know that the condition on the interpolation nodes $t_1,\ldots,t_m$ should look something like the ones appearing below

1) $min_{k,\ell}|t_k-t_\ell|\ge 1/n$

(with the distance meant to be circular that is |0.9-0.1|=0.2).

or

2) more sophisticated conditions like:

$D_{m}(t_1,\ldots,t_m)$ needs to be small. The discrepancy of a a finite sequence of real numbers $x_1,x_2,\ldots,x_N\in[0,1]$ is defined as \begin{equation*} D_N(x_1,x_2,\ldots,x_N)=\underset{0\le\alpha<\beta\le 1}{sup}\bigg|\frac{A([\alpha,\beta);N)}{N}-(\beta-\alpha)\bigg|, \end{equation*} with $A([\alpha,\beta);N)$ denoting the number of $x_i$ such that $x_i\in[\alpha,\beta)$ (Based on section 2 of Uniform Distribution of Sequences by Kuipers and Niederreiter).

Answer 1

This appears to be Proposition 2.1 from this paper. The bound they give (in Theorem 1.2 there) is $$\min_{k,l}|t_k-t_l| \geq 2/n, $$ the distance $|t_k-t_l|$ here is the distance between $t_k$ and $t_l$ on a torus, like a wrap-around distance, and they also assume $n\geq128$.

Answer 2

I now think that using the following should work \begin{equation*} \sum_k c_k{\bigg(\prod_{k\neq \ell}\frac{\sin \pi (t-t_\ell)}{\sin \pi (t_k-t_\ell)}\bigg)}^2 \end{equation*} I'm trying to see what's the best way to control each term in absolute value using the notion of discrepancy.

The reason I expect this to work is that the mentioned product decays fast enough like (1/t^2) so that when we sum them up it still remains less than one. Also when the $t_\ell$ are grid points $i/N$ the product becomes the Fejer kernel or the square of the Dirichlet kernel ${({\frac{\sin(\pi/N*t)}{N\sin(\pi*t)}})}^2$, which is always less than one. So we would be done if we can bound the difference between the Fejer kernel and the above kernels. Noting that if we where dealing with the grid points the answer would be the Fejer kernel.