Highly Nonlinear Approximations for Sparse Signal Representation
The spaces
,
and
The Euclidean space
is an inner product space with
inner product defined by
![$\displaystyle \langle {\mathbf{{x}}},{\mathbf{{y}}}\rangle = x_i \overline {y}_i + \ldots x_n \overline {y}_n,$](img124.png)
![$ {\mathbf{{x}}}=(x_1,\ldots,x_n)$](img125.png)
![$ {\mathbf{{y}}}=(y_1,\ldots,y_n)$](img126.png)
![$ \Vert{\mathbf{{x}}}\Vert$](img127.png)
![$\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}}.$](img128.png)
The space is an inner product space of
functions on
with
inner product defined by
![$\displaystyle \langle f,g \rangle = \int_a^b f(t) \overline {g}(t) dt$](img130.png)
![$\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}}.$](img131.png)
The space is the space of functions on
having continuous derivatives up to order
.
The space of continuous functions on
is denoted as
.