# Highly Nonlinear Approximations for Sparse Signal Representation

## Updating the oblique projector to

We assume that is known and write it in the explicit form

 (17)

In order to inductively construct the duals we have to discriminate two possibilities
i)
, i.e.,
ii)
, i.e.

Case i)

Proposition 4   Let and vectors in (17) be given. For an arbitrary vector the dual vectors computed as

 (18)

for and produce the identical oblique projector as the dual vectors .

Case ii)

Proposition 5   Let vector and vectors in (17) be given. Thus the dual vectors computed as

 (19)

where with , provide us with the oblique projector .

The proof these propositions are given in [10]. The codes for updating the dual vectors are FrInsert.m and FrInsertBlock.m.

Property 2   If vectors are linearly independent they are also biorthogonal to the dual vectors arising inductively from the recursive equation (19).

The proof of this property is in [10].

Remark 3   If vectors are not linearly independent the oblique projector is not unique. Indeed, if are dual vectors giving rise to then one can construct infinitely many duals as:

 (20)

where are arbitrary vectors in (see [10]).