memory bandwidth is less than standard gridding, in some situations.

In section \[sec:gauss\_wproj\] we describe Gaussian anti-aliasing functions used for $w$ projection and in section \[sec:wproj\] we apply these functions with the Fast Gauss Transform, to the gridding and degridding problems. In section \[sec:performance\] we derive the theoretical operations and memory requirements, and in section \[sec:perf\_comparison\] we compare the theoretical requirements for our method and the standard methods. In section \[sec:implementation\] we compare the theoretical results with a real-world implementation. We discuss our results in \[sec:discussion\] and we draw our conclusions in section \[sec:conclusions\].

$w$-projection with a Gaussian anti-aliasing function {#sec:gauss_wproj}
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In this section we describe the gridding process, and use a Gaussian anti-aliasing function to obtain a closed form expression for the convolution function.

We acknowledge that the anti-aliasing properties of a Gaussian are not ideal. Typical interferometric imaging uses prolate spheroidal wavefunctions, which have a optimal out-of-band rejection [@Schwab80]. Nonetheless, Gaussian functions have simple analytical relationships (e.g. the convolutions and Fourier transforms of Gaussians are both Gaussians) that allow us to proceed.

Gridding
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The aim of convolutional gridding is to take visibilities sampled on arbitrary $u,v,w$ coordinates and interpolate them onto a regular grid so that 