Highly Nonlinear Approximations for Sparse Signal Representation

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The spaces $ \mathbb{R}^n$, $ L^2[a,b]$ and $ C^k[a,b]$

The Euclidean space $ \mathbb{R}^n$ is an inner product space with inner product defined by

$\displaystyle \langle {\mathbf{{x}}},{\mathbf{{y}}}\rangle = x_i \overline {y}_i + \ldots x_n \overline {y}_n,$

with $ {\mathbf{{x}}}=(x_1,\ldots,x_n)$ and $ {\mathbf{{y}}}=(y_1,\ldots,y_n)$. The norm $ \Vert{\mathbf{{x}}}\Vert$ is induced by the inner product

$\displaystyle \Vert{\mathbf{{x}}}\Vert=\langle {\mathbf{{x}}},{\mathbf{{x}}}\ra... ... )^{\frac{1}{2}}= (\vert x_i\vert^2 + \ldots +\vert x_n\vert^2)^{\frac{1}{2}}.$

The space $ L^2[a,b]$ is an inner product space of functions on $ [a,b]$ with inner product defined by

$\displaystyle \langle f,g \rangle = \int_a^b f(t) \overline {g}(t) dt$

and norm

$\displaystyle \Vert f\Vert=\langle f,f \rangle ^{\frac{1}{2}}= \bigg(\int_a^b \vert f(t)\vert^2 dt\bigg)^{\frac{1}{2}}.$

The space $ C^k[a,b]$ is the space of functions on $ [a,b]$ having continuous derivatives up to order $ k\in \mathbb{N}$. The space of continuous functions on $ [a,b]$ is denoted as $ C^0[a,b]$.