We propose a new, general, method for spatial filtering by beam-forming. The desired filter, specified analytically on an n-dimensional sphere, is extended to n+1-dimensional Euclidean space. A continuous beamforming function is then obtained by the n+1-dimensional Fourier transform of the extended filter. The beamforming weight at a given array element corresponds then to a sample of the function at the array element location., The scheme is a generalisation of focused beamforming on a single point by phase difference alignment. The analytic framework allows tractable, stable determination of beamforming weights, and for clear filter specification. By avoiding approximating a Dirac, desired areas can be covered with reduced side lobes. Multiple areas may be targeted simultaneously. In communications applications, channel information updates can be reduced, and movement accounted for. A WiFi demonstration shows that more flexible beam-shapes can be beneficial for real-life examples, factoring in attenuation.