tokens: linear interpolation; see Fig. 2 . Based on these two cases, we expect that C ( u ) (and its derivatives) may be discontinuous at integer values of u when W is even, and may be discontinuous at half-integer values of u (i.e. values half-way between the integers) when W is odd. For W 3, visible discontinuities remain in the least-misfit gridding function C ( u ) at u 0.5 and u 1.5. This behaviour is a natural extension of the situation for W 1, and may also be understood by considering the set of three grid points on to which each visibility is distributed. According to which side of the mid-point between two grid points a baseline falls, the visibility is gridded on to a different set of three points. For example, if the baseline is u 0 0.49 (where u 0 is an integer), the visibility is distributed among u 0 1, u 0 , and u 0 1, whereas if the baseline is u 0 0.51, the visibility is distributed among u 0 , u 0 1, and u 0 2. Since the value of the gridding function at the edge of its support, i.e. C ( 1.5), is non-negligible, the way in which the visibility is distributed among the grid points is expected to change discontinuously as the baseline passes through u 0 0.5; this is because one grid point disappears from the set and is replaced by another. For W 4, these discontinuities in value and derivatives cease to be visible. In particular, the values of C ( u ) at the edge of the support u W /2 are very small. APPENDIX D: 34-SOURCE SIMULATED DATA Table D1 shows the location and flux of the 34 simulated point sources used in the numerical experiments. Sources are scattered about the phase centre with different fluxes across the full field of view. Table D1. Locations and fluxes of 34 simulated point sources. X and Y are specified in pixel numbers, and represent the distances from the image centre (0,0) to the corresponding point sources. Sources 27, 28, 29, and 30 are at four corners of the field, and are chosen to have the largest fluxes of the 34 sources. APPENDIX E: GENERATION OF THE LOOK-UP TABLE The look-up table for the least-misfit gridding function is assembled as follows: Create [0, 1) as a set of M s numbers spaced equally between 0 and 1. We refer to M s as the sampling rate in this paper. If W 11 then, for each value of , calculate a set of W gridding function values according to equation( 18 ) using matrix A , which can be used repeatedly as it is not dependent on . If W 11 then, for each value of , calculate a set of W gridding function values according to equation( B2 ) using matrix B . The B matrix must be recalculated for each , increasing the computational cost. APPENDIX F: EFFECT OF THE LOOK-UP TABLE Fig. 9 shows that, if the gridding function is calculated precisely, the RMS misfit between the DFT and the FFT is proportional to the square root of the map error ( x ), as expected from equation( 10 ). When, however, we store the gridding function C ( u ) for W /2 u W /2 in a table with WM s points over this range, and use
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