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

Abstract

We present an analysis of the approximation error for a $d$-dimensional quasiperiodic function $f$ with Diophantine frequencies, approximated by a periodic function with period $[0,L)^d$. When the $n$-dimensional ($n\geq d$) periodic function $F$ containing $f$ has certain regularity, the global behavior of $f$ can be described by a finite number $D$ Fourier components. The dominant part of periodic approximation error is bounded by $O(L^{-1/dD})$. Meanwhile, we discuss the optimal approximation rate. Finally, these analytical results are verified by some examples