  In this paper we translate to the cell-average setting the algorithm for the
point-value discretization presented in S. Amat, J. Ruiz, C.-W. Shu, D. F.
Y\'a\~nez, A new WENO-2r algorithm with progressive order of accuracy close to
discontinuities, submitted to SIAM J. Numer. Anal.. This new strategy tries to
improve the results of WENO-($2r-1$) algorithm close to the singularities,
resulting in an optimal order of accuracy at these zones. The main idea is to
modify the optimal weights so that they have a nonlinear expression that
depends on the position of the discontinuities. In this paper we study the
application of the new algorithm to signal processing using Harten's
multiresolution. Several numerical experiments are performed in order to
confirm the theoretical results obtained.
