# Highly Nonlinear Approximations for Sparse Signal Representation

## Downdating the oblique projector to

Let us suppose that by the elimination of the element the subspace is reduced to . In order to give the equations for adapting the corresponding dual vectors generating the oblique projector we need to consider two situations:
i)
i.e.,
ii)
, i.e.,

Case i)

Proposition 6   Let be given by (17) and let us assume that removing vector from the spanning set of leaves the identical subspace, i.e., . Hence, if the remaining dual vectors are modified as follows:

 (21)

the corresponding oblique projector does not change, i.e. .

Case ii)

Proposition 7   Let be given by (17) and let us assume that the vector to be removed from the spanning set of is not in . In order to produce the oblique projector the appropriate modification of the dual vectors can be achieved by means of the following equation

 (22)

The proof these propositions are given in [10]. The code for updating the vectors are FrDelete.m.