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

 such that (34)

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 .

The construction of nonuniform dictionaries arises from the next proposition [21]

Theorem 1   Suppose that are partitions of . Then