ON THE APPROXIMATION OF QUASIPERIODIC FUNCTIONS WITH DIOPHANTINE FREQUENCIES BY PERIODIC FUNCTIONS

We present an analysis of the approximation error for addimensional quasiperiodic function f with Diophantine frequencies, approximated by a periodic function with the fundamental domain [0, L1) \times [0, L2) \times \cdot \cdot \cdot \times [0, Ld). 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 the periodic approximation error is bounded by O(max1\leqj\leqd L-sj j ), where Lj belongs to the best simultaneous approximation sequence and sj is the number of different irrational elements in the jth dimension component of Fourier frequencies, respectively. Meanwhile, we discuss the approximation rate. Finally, these analytical results are verified by some examples.