On the approximation of quasiperiodic functions with Diophantine frequencies by periodic functions

We present an analysis of the approximation error for a $d$-dimensional quasiperiodic function $f$ with Diophantine frequencies, approximated by a periodic function with the fundamental domain $[0,L_1)\times [0,L_2)\times \cdots \times [0,L_d)$. When $f$ has a certain regularity, its global behavior can be described by a finite number of Fourier components and has a polynomial decay at infinity. The dominant part of periodic approximation error is bounded by $O({\max }_{1\le j\le d}{L}_j^{-s_j})$, where $L_j$ belongs to the best simultaneous approximation sequence and $s_j$ is the number of different irrational elements in $j$-th dimension component of Fourier frequencies, respectively. Meanwhile, we discuss the approximation rate. Finally, these analytical results are verified by some examples.