@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. }
}