projection algorithm.

Radially symmetric Gaussian anti-aliasing functions
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For a fixed $w$, equation \[eq:convfunc\] is the convolution of the anti-aliasing function with a complex Gaussian. If we choose a Gaussian for the anti-aliasing function $A$, the resulting convolution function is also Gaussian. To simplify notation, we will consider the radially symmetric problem by introducing a new parameter $t^2 = u^2 + v^2$, and can now write the Fresnel term as:

$$\begin{aligned}
G(t, w) & = & \frac{i}{w} \exp\left (- \frac{\pi i t^2}{w} \right ) \\
& = & \frac{i}{w} \exp \left ( \frac{-t^2}{ i \delta_G} \right )\end{aligned}$$

with $\delta_G = - w/ \pi$. We can also write the Gaussian anti-aliasing function as

$$A(t) = \exp{\left ( \frac{-t^2}{\delta_A}\right )}.$$

The gridding function is given by the convolution $G(t) * A(t)$, which is another Gaussian, whose variance is equal to the sum of the variances of the original Gaussians, i.e., to within a scaling factor:

$$\begin{aligned}
C(t, w) & = & A(t) * G(t, w) \\
& = & \frac{i}{w} \exp \left ( - t^2\frac{1}{\delta_{A} + i \delta_{G}} \right ) \\
& = & \frac{i}{w} \exp \left ( - t^2 \frac{\delta_{A}}{\delta^2_{A} + \delta^2_{G}} \right )
\exp \left (  -t^2 \frac{ - i 