In this work, we develop a DFT-based method for the interpolation of a real sequence that does not necessarily satisfy the Shannon-Whittaker sampling condition. Our derivation provides an insightful perspective to the interpolation problem and inspires us to formulate the interpolation as a signal recovery problem in the discrete frequency space. In combination with the downsampling operation and the Fourier shift theorem, the proposed interpolation scheme can be used to obtained regular samples of the function in question with arbitrary starting position and sampling density. All computations involved, in addition to FFT, are simple arithmetical operations; consequently, the proposed method is computationally efficient. Our computer experiments demonstrate that the proposed method can produce good interpolation results even when the functions are severely undersampled.