Singularities of Fourier transforms: A tribute to M. J. Lighthill (Q5926053)

This is a very personally written homage of M. J. Lightill and his theory of generalized functions. The author has extended earlier Lighthills theory in the one-dimensional case to the case of several variables. He considers expecially singular generalized functions, such as \(r^\beta\), \(r^\beta\log^mr\), \(r^\nu J_\nu(r)\), investigates their properties and calculates their Fourier transforms. It is proved that (under some conditions) the singularities of a generalized function determine the behaviour of its Fourier transform at infinity and the behaviour of a generalized function at infinity determines the singularities of its Fourier transform. Everyboty who is interested in this field will be pleased to read this paper.