Suppose we have given constants $A_i, x_i (i=1..N)$
Is it possible to approximately calculate the sum of N gaussians in less than N iterations for any x? (may be with some preprocessing)
$$\sum_{i=1}^{N}A_i e^{-(x-x_i)^2}$$
The same question for 2D case $$\sum_{i=1}^{N}A_i e^{-(x-x_i)^2-(y-y_i)^2}$$
If the $A_i$, $x_i$ and $y_i$ can take arbitrary values then it is hopeless because you even need $N$ iterations merely to read the input.
If (some of) the Gaussians have the same distribution then their sum can be expressed as a single Gaussian.