Optimal variance conjecture for Gaussian regularized Shannon sampling

Let $m$ be the truncation parameter, let $\beta$ and $\delta$ be the parameters occurring in the Shannon sampling formula, and let $E_1(m)$ and $E_2(m)$ denote its regularization and truncation errors. The Gaussian regularized Shannon sampling formula uses the Gaussian function with variance $4\sigma^2$ and has uniform approximation error $E(m)$ as in the cited formulas.

Optimal variance conjecture. The parameter specified by

$$is the optimal variance for the Shannon sampling formula$$

with the Gaussian function

, not only in the weak sense that $E_1(m)$ and $E_2(m)$ have the same exponential decay with respect to $m$, but also because it guarantees the maximum decay rate of the uniform approximation error

.

The weak optimality follows from the stated approximation theorem, whereas this stronger optimality has so far only been observed numerically and remains open.