The Fourier transform of an irreducible spherical tensor is normally computed with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. The angular integrations are then trivial. However, the remaining radial integral containing a spherical Bessel function may be so complicated that the applicability of Fourier transformation is severely restricted. As an alternative, the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested. The weakly convergent expansions of a plane wave are constructed in such a way that their application in Fourier integrals leads to expansions of the Fourier or inverse Fourier transform that converge with respect to the norm of either the Hilbert space L2(R3) or the Sobolev space W(1)2(R3). Accordingly, these weakly convergent expansions may be viewed as distributions that are defined on either L2(R3) or W(1)2(R3). The properties of some complete orthonormal sets of functions, in particular their Fourier transforms, are also studied. Shibuya and Wulfman [Proc. R. Soc. London Ser. A 286, 376 (1965)] derived an expansion of a plane wave involving the four-dimensional spherical harmonics. It is shown that this Shibuya–Wulfman expansion is also a distribution which is defined on the Sobolev space W(1)2(R3). Finally, as an application it is shown how weakly convergent expansions can be used profitably for the construction of addition theorems.