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

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