Vladikavkazskii Matematicheskii Zhurnal
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Vladikavkaz. Mat. Zh.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Vladikavkazskii Matematicheskii Zhurnal, 2020, Volume 22, Number 1, Pages 5–12
DOI: https://doi.org/10.23671/VNC.2020.1.57532
(Mi vmj710)
 

This article is cited in 1 scientific paper (total in 1 paper)

Three theorems on Vandermond matrices

A. E. Artisevicha, A. B. Shabatb

a Adyghe State University, 208 Pervomayskaya St., Maikop 385000, Russia
b Landau Institute for Theoretical Physics, 1A Akademika Semenova Ave., Chernogolovka 142432, Russia
Full-text PDF (222 kB) Citations (1)
References:
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 well-known problem of commuting symmetric Vandermonde matrices $\Lambda$ with (symmetric) three-diagonal 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 three-diagonal 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
Document Type: Article
UDC: 517.95
MSC: 42A38
Language: Russian
Citation: A. E. Artisevich, A. B. Shabat, “Three theorems on Vandermond matrices”, Vladikavkaz. Mat. Zh., 22:1 (2020), 5–12
Citation in format AMSBIB
\Bibitem{ArtSha20}
\by A.~E.~Artisevich, A.~B.~Shabat
\paper Three theorems on Vandermond matrices
\jour Vladikavkaz. Mat. Zh.
\yr 2020
\vol 22
\issue 1
\pages 5--12
\mathnet{http://mi.mathnet.ru/vmj710}
\crossref{https://doi.org/10.23671/VNC.2020.1.57532}
Linking options:
  • https://www.mathnet.ru/eng/vmj710
  • https://www.mathnet.ru/eng/vmj/v22/i1/p5
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Владикавказский математический журнал
    Statistics & downloads:
    Abstract page:209
    Full-text PDF :66
    References:29
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024