Using sine and cosine transforms to solve Poisson's equation with Dirichlet boundary conditions seem quite standard nowadays (see, e.g., here or Table 2 in this paper). In the case of Poisson's equation, the multiplication with $-k^2$ preserves the symmetry of the Fourier coefficients such that the same transform can be used for going back to real space.
But what happens if the differential operator in Fourier space changes the symmetry? E.g., in the case of the transport equation, multiplication with $ik$ would transform even symmetry into odd, such that another transform would be needed for the backtransformation to real space. This should happen for all first or mixed derivatives. Are there any papers/codes around for these cases?