  Quasiperiodic systems are important space-filling ordered structures, without
decay and translational invariance. How to solve quasiperiodic systems
accurately and efficiently is of great challenge. A useful approach, the
projection method (PM) [J. Comput. Phys., 256: 428, 2014], has been proposed to
compute quasiperiodic systems. Various studies have demonstrated that the PM is
an accurate and efficient method to solve quasiperiodic systems. However, there
is a lack of theoretical analysis of PM. In this paper, we present a rigorous
convergence analysis of the PM by establishing a mathematical framework of
quasiperiodic functions and their high-dimensional periodic functions. We also
give a theoretical analysis of quasiperiodic spectral method (QSM) based on
this framework. Results demonstrate that PM and QSM both have exponential
decay, and the QSM (PM) is a generalization of the periodic Fourier spectral
(pseudo-spectral) method. Then we analyze the computational complexity of PM
and QSM in calculating quasiperiodic systems. The PM can use fast Fourier
transform, while the QSM cannot. Moreover, we investigate the accuracy and
efficiency of PM, QSM and periodic approximation method in solving the linear
time-dependent quasiperiodic Schr\"{o}dinger equation.
