
This article is cited in 1 scientific paper (total in 1 paper)
Three theorems on Vandermond matrices
A. E. Artisevich^{a}, A. B. Shabat^{b} ^{a} Adyghe State University, 208 Pervomayskaya St., Maikop 385000, Russia
^{b} Landau Institute for Theoretical Physics, 1A Akademika Semenova Ave., Chernogolovka 142432, Russia
Abstract:
We consider algebraic questions related to the discrete Fourier transform defined using symmetric Vandermonde matrices $\Lambda$. The main attention in the first two theorems is given to the development of independent formulations of the size $N\times N$ of the matrix $\Lambda$ and explicit formulas for the elements of the matrix $\Lambda$ using the roots of the equation $\Lambda^N = 1$. The third theorem considers rational functions $f(\lambda)$, $\lambda\in \mathbb{C}$, satisfying the condition of “materiality” $f(\lambda)=f(\frac{1}{\lambda})$, on the entire complex plane and related to the wellknown problem of commuting symmetric Vandermonde matrices $\Lambda$ with (symmetric) threediagonal matrices $T$. It is shown that already the first few equations of commutation and the above condition of materiality determine the form of rational functions $f(\lambda)$ and the equations found for the elements of threediagonal matrices $T$ are independent of the order of $N$ commuting matrices. The obtained equations and the given examples allow us to hypothesize that the considered rational functions are a generalization of Chebyshev polynomials. In a sense, a similar, hypothesis was expressed recently published in “Teoreticheskaya i Matematicheskaya Fizika” by V. M. Bukhstaber et al., where applications of these generalizations are discussed in modern mathematical physics.
Key words:
Vandermond matrix, discrete Fourier transform, commutation conditions, Laurent polynomials.
Received: 16.07.2019
Citation:
A. E. Artisevich, A. B. Shabat, “Three theorems on Vandermond matrices”, Vladikavkaz. Mat. Zh., 22:1 (2020), 5–12
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https://www.mathnet.ru/eng/vmj710 https://www.mathnet.ru/eng/vmj/v22/i1/p5

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