# Highly Nonlinear Approximations for Sparse Signal Representation

#

Signal Representation, Reconstruction, and Projection

Regardless of its informational content,
in this tutorial we consider that a **signal**is an element of an inner product space with norm induced by the inner product, . Moreover, we assume that all the signals of interest belong to some finite dimensional subspace of spanned by the set . Hence, a signal can be expressed by a finite linear superposition

We call **measurement** or **sampling** to the
process of transforming a
signal into a number. Hence a **measure** or **sample** is a
*functional*.
Because we restrict considerations to
linear measures the associated functional is linear and
can be expressed as

for some

We refer the vector to as **measurement vector**.

Considering measurements , each of which is obtained by a measurement vector , we have a numerical representation of as given by

By denoting

where the operation indicates that acts by taking inner products, (1) is written as

**Subsections**