频率位于指定格点的平行六面体的指数基
Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice
AI总结：
该研究针对频率限制在指定格点的情形，推导了平行六面体存在指数Riesz基的两个充分条件，分别推广了正交基情形的充要条件、通过生成矩阵谱范数给出约束。
摘要：
自20世纪50年代以来，对于$\u211d^d$中给定平行六面体上支撑的平方可积函数空间，是否存在频率在$\u211d^d$中的傅里叶基的问题已得到充分研究。在一篇姊妹论文中，我们推导了$\u211d^d$中的平行六面体允许频率被限制为$\u211d^d$中指定格点子集的指数正交基的充要条件，这一限制在许多应用中具有相关性。本文研究了允许频率受相同约束的指数Riesz基的平行六面体的类似条件。我们给出了Riesz基情形下平行六面体的一个充分条件，该条件直接推广了正交基情形下得到的充要条件之一。我们还给出了一个充分条件，该条件约束生成平行六面体的矩阵的谱范数，而非约束矩阵的结构。
英文摘要：
The existence of a Fourier basis with frequencies in $\mathbb{R}^d$ for the space of square integrable functions supported on a given parallelepiped in $\mathbb{R}^d$, has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in $\mathbb{R}^d$ to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in $\mathbb{R}^d$, a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix.
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。
