You can get a very good approximation of the Gaussian function by convolving several box or simple-moving-average filters, in the same way that you get a good approximation of it with a spline, in a way that I don’t quite know how to name right now, but which is exactly equivalent to convolving several box filters.
From a computational perspective, notable features of this procedure include that box filters require no multiplications, require a very small number of operations per sample (only two additions and a subtraction, plus a FIFO buffer of the size of the box), and can be computed exactly with integer math in which the intermediate results overflow, as long as the final result does not. XXX is that even true? WTF
Precisely converting discretely sampled signals back into bandlimited continuous signal involves convolving them with the sinc function (the Fourier transform of the unit pulse) which is inconvenient because it decays very slowly, meaning that doing the convolution directly (in the spatial domain) requires summing weighted contributions from quite far away to get reasonable accuracy. If you just do a zero-order hold, turning the sampled curve into stairsteps, you are in effect convolving the modulated Dirac comb that is your discretely sampled signal with a unit pulse, thus filtering its frequency response with a sinc spectrum, which is disastrous for frequencies above about half Nyquist.
One approach to reconstruction filtering is to use Nth-order splines that pass through y=1 at x=0 and have zeroes at all other integers. (These are sometimes called cardinal B-splines, by analogy to the sinc “sinus cardinalis” function, but that terminology is arguably wrong and at least confusing.) These have very compact support, and they approach sinc in the limit where N → ∞. But they definitely require multiplication.
Derivatives of the Gaussian include the so-called Mexican Hat wavelet, Ricker wavelet, or Marr wavelet, which is its negated second derivative. This is vaguely close to sinc: it has a big round bulge in the middle with a zero on each side of it, then local minima, and then it asymptotically approaches zero as it goes to infinity.
I have this idea that the second derivative of the convolution of a signal with the Gaussian is equal to the convolution of the signal with the second derivative of the Gaussian, because both differentiation and convolution with something are linear time-invariant transforms. But now I’m not sure if that’s true.
The Mexican Hat Wavelet page on Wikipedia says, “In practice, this wavelet [in multiple dimensions] is sometimes approximated by the difference of Gaussians function, because the DoG is separable.” If this is true (it’s not backed up by the reference given) then it seems like a better way to convolve things with that wavelet in most cases would be to use the procedure I’ve outlined above to approximate the convolution using the Gaussian-then-differentiate approach. Convolving five box filters gives you a quartic approximation of a Gaussian; its second derivative is then a quadratic approximation of the Mexican Hat Wavelet. Note that this entire computation requires no multiplication.
Higher derivatives might also be useful, as they oscillate more times and hit more zeroes before dying out; they are the Hermite functions multiplied by the Gaussian. Their zeroes aren’t really in the right places, but that error might not be large enough to matter; H₄ has zeroes at ±sqrt((3±sqrt(6))/2)), which is about ±0.523 and ±1.650. That doesn’t oscillate nearly fast enough to be a good approximation for sinc. The higher-order Hermite polynomials’ roots have a similar problem: their zeroes are roughly evenly spaced, while sinc has an double wide space at x=0 where there would normally be a root.
These derivatives do, however, look very much like delicious zero-phase FIR narrow bandpass filter kernels, which are more or less like the Gabor kernel. So they may not be that useful for signal reconstruction, but maybe they could be very useful for narrow bandpass signal processing!
I thought maybe it was just as bad for reconstruction that the higher-order Gaussian derivatives vanished very quickly, but maybe that’s just an unavoidable artifact of a reasonable degree of locality, which is not an altogether bad thing.
(I got a lot of this information from http://bmia.bmt.tue.nl/people/bromeny/MICCAI2008/Materials/05%20Gaussian%20derivatives%20MMA6.pdf, which even mentions the use of Gaussian derivatives as bandpass kernel filters.)