A note on the eigenvalues of \(g\)-circulants (and of \(g\)-Toeplitz, \(g\)-Hankel matrices) (Q2017983)

A \(g\)-circulant matrix \(A\) of size \(n\times n\) is of the form \(A=[a_{(r-gs)\,\text{mod}\, n}]_{r,s=0}^{n-1}\), while a matrix \(A\) is called \(g\)-Toeplitz if \(A=[a_{r-gs}]_{r,s=0}^{n-1}\). The paper mainly studies the eigenvalues of \(g\)-circulants and eigenvalue distribution of \(g\)-Teoplitz sequence. Closed form expressions of the eigenvalues of \(g\)-circulants are provided. If the entries \(a_k\) are Fourier coefficients of an integrable funcion \(f\) over the domain \((-\pi, \pi)\), a preliminary analysis of the spectral distribution of \(g\)-Toeplitz sequence is considered and the results are given in Theorem 5.1 and 5.2. A conjecture is given.