The spectral reconstruction theory of the reconstructive spectrometer is applied to the Fourier transform spectrometer. The method takes advantage of the high spectral resolution of the reconstructive spectrometer and the inherent high incident optical throughput of the Fourier transform spectrometer. Verification experiments were performed in the wavelength range of 520-530 nm with a simple structured experimental setup of a spatial heterodyne Fourier transform spectrometer. We used the collected pattern images of input beams with different single wavelengths for spectral calibration experiments. The spectrometer can realize unique one-to-one mapping between the patterns and wavelengths required by the spectral reconstruction theory. Spectral reconstruction using the spectral calibration patterns realizes a spectral resolution of 0.10 nm, a considerable improvement from ~5.65 nm obtained using the Fourier transform spectrometer principle. Finally, an additional pattern image of the beam with a 525 nm wavelength was used in the spectral reconstruction experiment. The reconstructed spectrum contains reconstruction errors, and the full width at half maximum (FWHM) of the spectral signal peak is ~0.30 nm. The correlation analysis of the pattern images shows that spectral reconstruction is affected by noise during pattern image collection and the high similarity of the pattern images of the incident beams of adjacent wavelengths. Nevertheless, the reconstructed spectrum reveals the spectral information of the incident light. Moreover, the smaller FWHM spectral signal peak than that obtained by the Fourier transform spectrometer verifies the feasibility and the advantage of high spectral resolution of applying the spectral reconstruction theory to the Fourier transform spectrometer.