什么是有限傅里叶变换的真谱
What is a true spectra of a finite Fourier transform
摘要：
本文讨论有限交换群$G$和抽象傅里叶变换${\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$ .然后，我们考虑$\tilde{j}\circ {\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$，其中$\tilde j:{\mathbb C}^\hat{G}\to {\mathbb C}^G$由具有双射$j:G\to \hat{G}$的复合定义。（$\tilde j$是$j$的回撤。）特别地，我们证明了$(\tilde{j}\circ {\mathcal F})^2$是置换当且仅当$j$是群同构.然后，我们研究了$\tilde{j}\circ{\mathcal F}$的光谱如何依赖于同晶$j$。
英文摘要：
In this paper we deal with a finite abelian group $G$ and the abstract Fourier transform ${\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$. Then, we consider $\tilde{j}\circ {\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$ where $\tilde j:{\mathbb C}^\hat{G}\to {\mathbb C}^G$ is defined by the composition with a bijection $j:G\to \hat{G}$. ($\tilde j$ is a pullback of $j$.) In particular, we show that $(\tilde{j}\circ {\mathcal F})^2$ is a permutation if and only if $j$ is a group isomorphism. Then, we study how the spectra of $\tilde{j}\circ{\mathcal F}$ depends on the isomorphism $j$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
