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tokens: Fourier transform of a Gaussian function is also Gaussian, and leads tohowever, they comment that this expression breaks down at large FoVs and diverges at w = 0. By choosing to fix the sky to a parabola, rather than the sphere, we arrive at the same approximation above. First, we choosethen by integrating over n in Equation ( 14 ), we arrive at the same small FoV approximation. 3.2. <italic>w</italic>-projection with Exact Spherical Correction We choose to replace the horizon with a window function, where the expression for the full sphere isAny scaling from this window function can be corrected in the upper hemisphere of the measurement equation 3.2.1. No Horizon When h ( n ) = 1 there is no horizon and the w -projection kernel is calculated fromThe Fourier transform of this equation has an analytic solution that can be simply expressed as a real valued functionas shown in Vembu ( 1961 ), which is solved in spherical coordinates due to symmetry. This solution dates back as far as Poisson ( 1820 ), and similar problems have been solved in two-dimensions in Parseval ( 1805 ). The units of ( u , v , w ) are implicitly chosen to depend on the directional cosines ( l , m , n ), meaning corresponds to the largest spatial scales. The Sinc function above represents limits on the resolution in ( u , v , w ) due to the FoV being bounded to the sphere. The uncertainty principle states that restricting the FoV is equivalent to enforcing a resolution limit on C ( u , v , w ). At a small FoV, this kernel is effectively a delta function of small support. However, as the FoV increases, the kernel becomes a radial Sinc function with extended support and rapid oscillations. When mosaicking multiple FoVs, the resolution in ( u , v , w ) is increased (as discussed in Ekers Rots 1979 and Thompson 1999 ); however, the total FoV will be limited to the sphere, as represented by this radial Sinc function. Since x ( l , m ) is independent of n it will project both onto the sphere for n and − n . While C ( u , v , w ) models the curvature of the sphere, it allows a reflection of x ( l , m ) for −1 ≤ n 0. This is why a horizon window function needs to be included in the analysis. 3.2.2. Projecting above the Horizon If we let H ( w ) be the Fourier transform of h ( n ), we find that the horizon effect can be understood through the convolution theorem:We can get an expression for the horizon-limited w -projection kernel in the ( u , v , w ) domain in terms of the w -projection kernel for the full sphere. Choosing h ( n ) = Θ( n ) with , we find an expression equivalent to Equation ( 17 ) in the ( u , v , w ) domainwhere the second term is a Hilbert transform of the sphere along the w -axis. Another equivalent expression can be found by choosing a box function for the horizon window, by setting ,We are not aware of an analytic solution to this convolution, which could improve understanding of the behavior of wide-field effects. 3.3. Convolution with a Gridding Kernel To