Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Principal Non-singularity of Fourier Matrices on $\mathbb Z_p \times \mathbb Z_q$ and $\mathbb Z_2^k \times \mathbb Z_q$

Weiqi Zhou

Abstract

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.

Principal Non-singularity of Fourier Matrices on $\mathbb Z_p \times \mathbb Z_q$ and $\mathbb Z_2^k \times \mathbb Z_q$

Abstract

Let be the Fourier matrix on the cyclic group , a renowned theorem of Chebotarëv asserts that all minors in for prime are non-zero. In this short note it is shown that (i) all principal minors in the Kronecker product are non-vanishing (principal non-singularity) for distinct odd primes if is large enough and generates the multiplicative group ; (ii) the Fourier matrix on is principally non-singular upon permutation (in particular, for the identity permutation suffices) for odd prime and . 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.
Paper Structure (2 sections, 8 theorems, 16 equations)

This paper contains 2 sections, 8 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Lemma 1

frenkel2003 Let $D$ be an integral domain of characteristic $q$, and $p(x)\in D[x]$ a polynomial. Suppose that $a\neq 0$ is a root of $p(x)$ with multiplicity $m$, and $h$ is the number of non-zero coefficients of $p(x)$, then $m<h$ holds if one of the following two conditions holds:

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Corollary 1
  • proof
  • Lemma 5
  • ...and 2 more