# Highly Nonlinear Approximations for Sparse Signal Representation

## Projectors

An operator
is a
projector if it is *idempotent*, i.e.,

such that

Thus, if
,

such that

It is clear then that to reconstruct a signal
by means of
(1) the involved measurement vectors
,
that we shall also called henceforth **duals**, should give rise to an operator of the form (2), which must be a projector onto . Notice that the required operator is not unique, because there exist many projectors onto having different . Thus, for reconstructing signals in the range of the projector its null space can be chosen arbitrarily. However, the null space becomes extremely important when the projector acts on signals outside its range. A popular projector is the orthogonal one.