pixel and $uv$-cell interchangeably. To convert from wavelengths to pixels, a size of $uv$-cell is required. The size of the $uv$-cell, (in wavelengths) is set by the inverse of the desired field of view (in radians) i.e. $\phi = 1/\theta_{\rm f.o.v}$.

Therefore, a Gaussian width in pixels can be calculated from the width in wavelength by:

$$\delta (\rm pixels) = \frac{\delta (\rm wavelengths)}{\phi^2}$$

$w$ projection with the Fast Gauss Transform {#sec:wproj}
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A brief introduction to the Fast Gauss Transform
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In this section we introduce the Fast Gauss Transform (FGT) with variable scales [@Strain91] (hereafter S91), which is an approximate technique for calculating the sum of Gaussians. We conform to the language of S91, which uses the term ‘source’ to describe a Gaussian function with a given position, amplitude and width. We will describe the application of the FGT onto the gridding problem in section \[sec:gridding\].

The FGT works by partitioning the evaluation region into square boxes, with each box containing a set of Taylor coefficients (Fig. \[fig:grid\]). For each Gaussian ‘source’, the Taylor coefficients for the boxes surrounding the source position are updated, with the size of the updated region being defined by the ‘width’ of the source, 