On the approximation of quasiperiodic functions with Diophantine frequencies by periodic functions
arxiv(2023)
摘要
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 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(L^-1/s), where L belongs to the best simultaneous
approximation sequence of Fourier frequencies and s is the number of
different irrational elements in Fourier frequencies. Meanwhile, we discuss the
optimal approximation rate. Finally, these analytical results are verified by
some examples.
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