Principal Non-singularity of Fourier Matrices on $\mathbb Z_p \times \mathbb Z_q$ and $\mathbb Z_2^k \times \mathbb Z_q$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911473151770624 |
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| author | Zhou, Weiqi |
| author_facet | Zhou, Weiqi |
| contents | Let $F_n$ be the $n\times n$ Fourier matrix on the cyclic group $\mathbb Z_n$, a renowned theorem of Chebotarëv asserts that all minors in $F_n$ for prime $n$ are non-zero. In this short note it is shown that (i) all principal minors in the Kronecker product $F_p\otimes F_q$ are non-vanishing (principal non-singularity) for distinct odd primes $p,q$ if $q$ is large enough and generates the multiplicative group $\mathbb Z_p^*$; (ii) the Fourier matrix on $\mathbb Z_2^k \times \mathbb Z_q$ is principally non-singular upon permutation (in particular, for $k=1$ the identity permutation suffices) for odd prime $q$ and $k=1,2,3$. The proof is just an exposition of existing techniques reorganized in a unified way. The result will have implications in combining Riesz bases of exponentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Principal Non-singularity of Fourier Matrices on $\mathbb Z_p \times \mathbb Z_q$ and $\mathbb Z_2^k \times \mathbb Z_q$ Zhou, Weiqi Classical Analysis and ODEs 42A99, 15A15 Let $F_n$ be the $n\times n$ Fourier matrix on the cyclic group $\mathbb Z_n$, a renowned theorem of Chebotarëv asserts that all minors in $F_n$ for prime $n$ are non-zero. In this short note it is shown that (i) all principal minors in the Kronecker product $F_p\otimes F_q$ are non-vanishing (principal non-singularity) for distinct odd primes $p,q$ if $q$ is large enough and generates the multiplicative group $\mathbb Z_p^*$; (ii) the Fourier matrix on $\mathbb Z_2^k \times \mathbb Z_q$ is principally non-singular upon permutation (in particular, for $k=1$ the identity permutation suffices) for odd prime $q$ and $k=1,2,3$. The proof is just an exposition of existing techniques reorganized in a unified way. The result will have implications in combining Riesz bases of exponentials. |
| title | Principal Non-singularity of Fourier Matrices on $\mathbb Z_p \times \mathbb Z_q$ and $\mathbb Z_2^k \times \mathbb Z_q$ |
| topic | Classical Analysis and ODEs 42A99, 15A15 |
| url | https://arxiv.org/abs/2505.01189 |