  Quasiperiodic systems, related to irrational numbers, are space-filling
structures without decay nor translation invariance. How to accurately recover
these systems, especially for non-smooth cases, presents a big challenge in
numerical computation. In this paper, we propose a new algorithm, finite points
recovery (FPR) method, which is available for both smooth and non-smooth cases,
to address this challenge. The FPR method first establishes a homomorphism
between the lower-dimensional definition domain of the quasiperiodic function
and the higher-dimensional torus, then recovers the global quasiperiodic system
by employing interpolation technique with finite points in the definition
domain without dimensional lifting. Furthermore, we develop accurate and
efficient strategies of selecting finite points according to the arithmetic
properties of irrational numbers. The corresponding mathematical theory,
convergence analysis, and computational complexity analysis on choosing finite
points are presented. Numerical experiments demonstrate the effectiveness and
superiority of FPR approach in recovering both smooth quasiperiodic functions
and piecewise constant Fibonacci quasicrystals. While existing spectral methods
encounter difficulties in accurately recovering non-smooth quasiperiodic
functions.
