I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of complex analysis is limited; essentially I know a subset of what is in Rudin's "Real and Complex Analysis".

Precisely, I would like to know if there is a coefficient sequence $(c_\ell)_{\ell \in \Bbb{Z}} \subset \Bbb{C}$ satisfying the following:

1) $\sum_{\ell \in \Bbb{Z}} (e^{\ell^2} \cdot |c_\ell|)^2 < \infty$;

2) The Laurent series $\varphi(z) := \sum_{\ell \in \Bbb{Z}} c_\ell z^\ell$ satisfies $\varphi(e^{2n}) = 0$ for all $n \in \Bbb{N}_0 = \{0,1,2,\dots\}$.

If the Laurent series was actually a power series, then I know Jensen's formula (see for instance [Rudin, Real and Complex Analysis, Theorem 15.20]) which shows that if $f : \Bbb{C} \to \Bbb{C}$ is holomorphic, then $$ M(2r) \geq \prod_{n=1}^{n(2r)} \frac{2r}{|\alpha_n|}, $$ where the $\alpha_i$ are the zeroes of $f$, and where $|\alpha_1| \leq |\alpha_2| \leq \dots$, and where $n(2r)$ is the number of zeroes of $f$ satisfying $|z| < 2r$.

But even this estimate does not seem to rule out the existence of such a function: For $r = e^{2N}$, the RHS of Jensen's inequality is $2^N e^{N^2 - N}$. Conversely, given my assumptions on the decay of the coefficients, I can estimate $M(2r) \lesssim \sum_{\ell \in \Bbb{Z}} e^{-\ell^2} (2 e^{2N})^\ell$, where I think the estimate is quite sharp, and where the $N$-th term is precisely $(2 e^{N})^N$, so that I don't seem to get a contradiction to Jensen's estimate.

Finally, I also know that there are results about the existence of holomorphic functions with prescribed zeros, but I don't know of any result that gives decent control over the coefficients of the power (or Laurent) series.

Any help would be greatly apprechiated.