# Highly Nonlinear Approximations for Sparse Signal Representation

## Background and notations

We refer to the fundamental books [23,25] for a complete treatment of splines. Here we simply introduce the notation and basic definitions which are needed in the present context.

**Definition 1**

*Given a finite closed interval we define a*

**partition**of as the finite set of points*We further define subintervals as: and .*

**Definition 2**

*Let be the space of polynomials of degree smaller than or equal to , i.e.,*

*and define*

(35) |

*where indicates the restriction of the function on the interval .*

**Theorem 1**

*Suppose that are partitions of . Then*