@TechReport{ it:2017-010,
author = {Sven-Erik Ekstr{\"o}m and Stefano Serra-Capizzano},
title = {Eigenvalues and Eigenvectors of Banded {T}oeplitz Matrices
and the Related Symbols},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2017},
number = {2017-010},
month = may,
abstract = {It is known that for the tridiagonal Toeplitz matrix,
having the main diagonal with constant $a_0=2$ and the two
first off-diagonals with constants $a_{1}=-1$ (lower) and
$a_{-1}=-1$ (upper), there exists closed form formulas,
giving the eigenvalues of the matrix and a set of
associated eigenvectors. The latter matrix corresponds to
the well known case of the 1D discrete Laplacian, but with
a little care the formulas can be generalized to any triple
$(a_0,a_{1},a_{-1})$ of complex values.
In the first part of this article, we consider a
tridiagonal Toeplitz matrix of the same form
$(a_0,a_{\omega},a_{-\omega})$, but where the two
off-diagonals are positioned $\omega$ steps from the main
diagonal instead of only one. We show that its eigenvalues
and eigenvectors also can be identified in closed form. To
achieve this, ad hoc sampling grids have to be considered,
in connection with a new symbol associated with the
standard Toeplitz generating function. In the second part,
we restrict our attention to the symmetric real case
($a_0,a_{\omega}=a_{-\omega}$ real values) and we analyze
the relations with the standard generating function of the
Toeplitz matrix. Furthermore, as numerical evidences
clearly suggest, it turns out that the eigenvalue behavior
of a general banded symmetric Toeplitz matrix with real
entries can be described qualitatively in terms of that of
the symmetrically sparse tridiagonal case with real $a_0$,
$a_{\omega}=a_{-\omega}$, $\omega=2,3,\ldots$, and also
quantitatively in terms of that having monotone symbols, as
those related to classical Finite Difference discretization
of the operators $(-1)^q \frac{\partial^{2q}}{\partial
x^{2q}}$, where the case of $q=1$ coincides with $a_0=2$,
$a_{1}=a_{-1}=-1$.}
}