Quasi-inverses and approximation with min-max operators in the ℓ₁-norm

Abstract

The semi-group of min-max operators, as used for nonlinear smoothing or multiresolution analysis, has no nontrivial inverses. Having chosen a smoother for a specific purpose, the secondary approximation problem of minimising damage was considered by showing that quasi-inverses exist. This was done with respect to the total variation as norm in ℓ₁, as this is natural for these operators. We show that these quasi-inverses also minimise the residual in the more usual 1-norm.