Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Question

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).

Answer 1

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 $t_k$).