  We propose a spectral viewpoint for grain boundaries that are generally
quasiperiodic. To accurately capture the spectra computationally, it is crucial
to adopt the projection method for quasiperiodic functions. Armed with the
Lifshitz-Petrich free energy, we take the spectral viewpoint to examine tilt
grain boundaries of the hexagonal phase. Several ingredients of grain
boundaries are extracted, which are not easy to obtain from real-space
profiles. We find that only a few spectra substantially contribute to the
formation of grain boundaries. Their linear relation to the intrinsic spectra
of the bulk hexagonal phase is independent of the tilt angle. By examining the
feature of the spectral intensities, we propose a definition of the interface
width. The widths calculated from this definition are consistent with visual
estimation.
