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

$\displaystyle \Delta :=\{x_i\}_{i=0}^{N+1}, N\in\mathbb{N},$ such that $\displaystyle c=x_0<x_1<\cdots<x_{N}<x_{N+1}=d.$

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We further define subintervals $I_i, i=0,\dots,N$ as: $I_i=[x_i,x_{i+1}), i=0,\dots,N-1$ and $I_N=[x_N,x_{N+1}]$ .

Definition 2 Let $\Pi_{m}$ be the space of polynomials of degree smaller than or equal to $m\in\mathbb{N}_0=\mathbb{N}\cup\{0\}$ , i.e.,

$\displaystyle \{\Pi_m(x): \Pi_m(x)=\sum_{i=0}^m a_ix^i,\; x\in \mathbb{R}\Big\},$

and define

$\displaystyle S_m(\Delta )=\{f\in C^{m-2}[c,d] : f\vert _{I_i}\in\Pi_{m-1}, i=0,\dots,N\},$

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where $f\vert _{I_i}$ indicates the restriction of the function on the interval ${I_i}$ .

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

Theorem 1 Suppose that $\Delta_j,j=1,\ldots,n$ are partitions of . Then

$\displaystyle S_m(\Delta_1) +\cdots + S_m(\Delta_n) = S_m(\cup_{j=1}^n\Delta_j).$

Highly Nonlinear Approximations for Sparse Signal Representation

Background and notations