论文标题
有限循环组的短时傅立叶变换的不确定性原理：平等案例
The uncertainty principle for the short-time Fourier transform on finite cyclic groups: cases of equality
论文作者
论文摘要
循环组$ \ mathbb {z} _n $上的不确定性原理的众所周知版本指出，对于任何几个函数，$ f，g \ in \ ell^2（\ mathbb {z} _n）\ setMinus \ setMinus \ {0 \} $该结果可以被视为$ \ mathbb {z} _n $上著名的Donoho-Stark不确定性原理的时频版本。但是，与Donoho-Stark原则不同，仍然缺少对极端物质的完整识别。在本说明中，我们通过证明$ v_g f $的支持具有基数$ n $，并且只有当时是$ \ mathbb {z} _n \ times \ times \ atmathbb {z} _n $时，我们才能为此问题提供答案。另外，我们完全确定相应的极端功能$ f，g $。除了翻译和调制外，问题的对称性是由与$ {\ rm sl}元素相关的某些元容器进行编码的（2，\ mathbb {z} _ {n/a}）$，其中$ a $是$ n $的$ a $。对有限的阿贝尔群体进行了部分概括。
A well-known version of the uncertainty principle on the cyclic group $\mathbb{Z}_N$ states that for any couple of functions $f,g\in\ell^2(\mathbb{Z}_N)\setminus\{0\}$, the short-time Fourier transform $V_g f$ has support of cardinality at least $N$. This result can be regarded as a time-frequency version of the celebrated Donoho-Stark uncertainty principle on $\mathbb{Z}_N$. Unlike the Donoho-Stark principle, however, a complete identification of the extremals is still missing. In this note we provide an answer to this problem by proving that the support of $V_g f$ has cardinality $N$ if and only if it is a coset of a subgroup of order $N$ of $\mathbb{Z}_N\times \mathbb{Z}_N$. Also, we completely identify the corresponding extremal functions $f,g$. Besides translations and modulations, the symmetries of the problem are encoded by certain metaplectic operators associated with elements of ${\rm SL}(2,\mathbb{Z}_{N/a})$, where $a$ is a divisor of $N$. Partial generalizations are given to finite Abelian groups.
