# Highly Nonlinear Approximations for Sparse Signal Representation

## Updating the oblique projector to

We assume that is known and write it in the explicit formIn 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*

*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*

*where with , provide us with the oblique projector .*

**Property 2**

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