A generalisation of Slepian's solution for the singular value decomposition of filtered Fourier transforms

It is shown that the Slepian method of finding the singular value decomposition for a band-limited Fourier transform can be generalised to more general filters. Assuming the filters to be spherically symmetric and satisfying a symmetry condition which is called 'Slepian symmetry', the most general case is found where a differential operator of the form D=- Del .( alpha (q) Del )+U(q) commutes with the filtered Fourier transform in R^d. The Slepian solution and a solution by Gori (1985) are contained as special cases.