Fourier Transform Spectrometer(FTS) has been extensively utlized in solar activity observation because of its ultra-high spectral resolution. During the process of spectral reconstruction using zero-crossing single sided interferogram, spectral are subject to frequency leakage, side lobes effects, and reuse of sampled data near the zero optical path difference resulting in spectral smoothing and distortion. In order to obtain more accurate solar spectra, it is necessary to apply weighted apodization operations to the zero-crossing single sided interferogram. An odd polynomial asymmetric weighted apodization function is utilized to carry out weighted and apodization operations on the zero-crossing single-sided interferogram. An investigation is conducted to assess the impact of employing diverse apodization functions in the construction of an asymmetric weighted apodization function for spectral reconstruction purposes. The simulated experimental results demonstrate that the use of odd polynomial asymmetric weighted apodization function effectively facilitates spectral recovery. Additionally, when compared with the weighted apodization methods proposed by Griffiths, Xiang Libin, and Sumner P. Davis, the method proposed exhibits a smaller error. Finally, the proposed method is further applied to the interferogram obtained from the Bruker IFS-125HR . Compared to other apodization methods, the use of the method proposed in this paper yields superior results in solar spectral recovery.