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).



Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay
Some initial thoughts to get us started:


Consider
$$
g(z) = \bigg ( \frac{\sin(2\pi(z - t\_0))}{2\pi(z - t\_0)} \cdot \prod\_{k \neq 0} \frac{\sin(2\pi(z - t\_k))}{\sin(2\pi(t\_0 - t\_k))} \bigg )^2
$$
Note that the Fourier transform of $g$ is compactly supported (by Paley-Wiener).
Now define
$$
h(x) := \sum\_{n \in \mathbb{Z}} g(x+n)
$$
Note that $h(t\_0)=1$ and $h(t\_k)=0$. Now apply Poisson summation and get
$$
h(x) = \sum\_{n \in \mathbb{Z}} \widehat{g}(n) e(x n)
$$
Note that this is a finite sum because $\widehat{g}$ is compactly supported.
Now also notice that $h$ decays like $\ll 1/(1+||z - t\_0||)^2$ where $|| \cdot ||$ denotes your circular distance (I haven't checked carefully). The decay can be increased to an arbitrary power by taking higher powers in the definition of $g$ but the price to pay is larger support of $\widehat{g}$. Finding the smallest support of $\widehat{g}$ could possibly be a research level problem. I would suggest looking at Valeer survey on extremal functions and/or Chapter 1 of Montgomery's book "Ten lectures on the interface between analytic number theory and harmonic analysis".


EDIT: Try also looking for Levin's book "Lectures on entire functions". There is a chapter on interpolation and sine-like functions. This will be useful for another question you asked about the product over the $t\_k$'s. Also take a look at some paper of Boas, since he was also concerned with product of the type that appear above (the product over $t\_k$).


