Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form \begin{equation*} f(t)=\sum_{k=-n}^n c_k e^{2\pi k it} \end{equation*}

(with n as small as possible) such that

1) $f(t_0)=1$, and $f(t_k)=0$ for $k\ge1$.

2) $|f(t)|\le \frac{c}{1+|t-t_0|^2}$. That is, $f(t)$ decays quickly away from $t_0$. Here, the distance $|t-t_0|$ is a circular distance ($|0.1-0.9|=0.2$).

I know that a condition on the interpolation nodes $t_0,t_1,\ldots,t_m$ like the ones appearing below is required

1) smallest constant $c$ such that

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

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

or

2) more sophisticated conditions like:

$D_{m+1}(t_0,t_1,\ldots,t_m)$ 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).