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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2304.04334 |
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| _version_ | 1866909380967923712 |
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| author | Jiang, Kai Li, Shifeng Zhang, Pingwen |
| author_facet | Jiang, Kai Li, Shifeng Zhang, Pingwen |
| contents | 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 $O(\max_{1\leq j \leq d} L_j^{-s_j})$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the approximation of quasiperiodic functions with Diophantine frequencies by periodic functions Jiang, Kai Li, Shifeng Zhang, Pingwen Number Theory 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 $O(\max_{1\leq j \leq d} L_j^{-s_j})$, 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. |
| title | On the approximation of quasiperiodic functions with Diophantine frequencies by periodic functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2304.04334 |