@TechReport{ it:2021-002,
author = {Rafael Diaz Fuentes and Mariarosa Mazza and Stefano
Serra-Capizzano},
title = {A {$\omega$}-Circulant Regularization for Linear Systems
Arising in Interpolation with Subdivision Schemes},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2021},
number = {2021-002},
month = mar,
abstract = {In the curve interpolation with primal and dual form of
stationary subdivision schemes, the computation of the
relevant parameters amounts in solving special banded
circulant linear systems, whose coefficients are related to
quantities arising from the used stationary subdivision
schemes. In some important cases it happens that the
associated generating function, which is a special Laurent
polynomial called symbol, has zeros on the unit complex
circle of the form exp$(2\pi \i j/n)$, where $n$ is the
size of the matrix, $\i^2=-1$, and $j$ is a non-negative
integer bounded by $n-1$. When this situation occurs the
discrete problem is ill-posed simply because the circulant
coefficient matrix is singular and the problem has no
solution (or infinitely many). Standard and nonstandard
regularization techniques such as least square solutions or
Tikhonov regularization have been tried, but the quality of
the solution is not good enough. In this work we propose a
structure preserving regularization in which the circulant
matrix is replaced by the $\omega$-circulant counterpart,
with $\omega$ being a complex parameter. A careful choice
of $\omega$ close to $1$ (recall that the set of
$1$-circulants coincides with standard circulant matrices)
allows to solve satisfactorily the problem of the
ill-posedness, even if the quality of the reconstruction is
reasonable only in a restricted number of cases. Numerical
experiments and further algorithmic proposals are presented
and critically discussed. }
}