where $t_C$ is the position of the $uv$ cell, and $\beta$ runs from $0$ to $p$.

The error in the evaluation of the sum is bounded by: $$E_p   \le  V \frac{r^p_p}{1 - r_p},$$

where $$\begin{aligned}
V = \sum_{j=1}^{M} |v_j|.\end{aligned}$$

$r$ and $p$ are chosen to make $|E_p| \le V \epsilon$.

Extension to complex Gaussians
------------------------------

The formulation of S91 is explicitly for real Gaussians, i.e. with $\delta_j, v_j \in \Re$, but in our case $\delta_j, v_j \in \mathds{C}$. A full derivation of the algorithm for complex values is outside the scope of this work . The important questions are whether the series of Equation \[eq:cbeta\] converges (and for what values of a complex $\delta$) and what is a suitable range ($R_j$).

Empirically, we have determined that:

1.  Using a range of $R_j = \sqrt{-\delta_{R,j} \log \epsilon}$ provides good results. This also makes sense because it is the amplitude of the envelope that determines the range.

2.  For calculating the order $q$ (Equation \[eq:qerror\]) we used $$\delta_j = max(\delta_A, \delta_I) \label{eq:delta_error_complex}$$

    Other possibilities (e.g. $\delta_j = \delta_R$) reduced $q$ too aggressively as $w$ increased, which resulted in very large errors.

In our case, the minimum value of $\delta$ occurs when $w=0$ 