# Highly Nonlinear Approximations for Sparse Signal Representation

### Example 1

Let us assume that the signal processing task is to approximate a signal by a signal . In this case, one normally would choose because this is the unique signal in minimizing the distance . Indeed, let us take another signal in and write it as . Since is orthogonal to every signal in we have

Hence is minimized if .

Remark 1   Any other projection would yield a distance which satisfies [2]

where is the minimum angle between the subspaces and . The equality is attained for , which corresponds to the orthogonal projection.