There are few papers where Fourier transform is used on formulating lifting-surface theories. Application of Fourier transform however has many merits over other methods, and is essential for some special problems. In supersonic flows, spanwise Fourier transform is applicable withoutdifficulties. In subsonic flows both spanwise and streamwise Fourier transforms are used, but disturbances infinitely upstream never vanish. This is a sort of paradox which must be circumvented. If the disturbances infinitely upstream are merely forced to vanish against mathematical logic, then the correct formulations have been attained for all cases investigated.