Highly Nonlinear Approximations for Sparse Signal Representation
Linear operators and linear functionals
Let
and
be vectors spaces.
A mapping
is a
linear operator if
![$\displaystyle \hat{A}(v + w) = \hat{A}v + \hat{A}w,\quad \hat{A}(av)= a \hat{A}v,$](img68.png)
![$ v, w \in {\cal{V}}_1$](img69.png)
![$ a\in F$](img70.png)
![$ {\cal{V}}_1$](img57.png)
![$ \hat{A}$](img71.png)
![$ {\cal{V}}_2$](img58.png)
![$ {\cal{V}}_1$](img57.png)
![$ {\cal{V}}_1$](img57.png)
![$ {\cal{V}}_1$](img57.png)
The adjoint of an operator
is the unique operator
satisfying that
![$\displaystyle \langle \hat{A} g_1, g_2\rangle = \langle g_1, \hat{A}^\ast g_2\rangle .$](img73.png)
![$ \hat{A}^\ast= \hat{A}$](img74.png)
An operator
has an inverse if there exists
such that
![$\displaystyle \hat{A}^{-1} \hat A =\hat{I}_{{\cal{V}}_1}$](img76.png)
![$\displaystyle \quad
\hat A \hat{A}^{-1} =\hat{I}_{{\cal{V}}_2},$](img77.png)
![$ \hat{I}_{{\cal{V}}_1}$](img78.png)
![$ \hat{I}_{{\cal{V}}_2}$](img79.png)
![$ {{\cal{V}}_1}$](img80.png)
![$ {{\cal{V}}_2}$](img81.png)
![$ \hat{A}^{\dagger}$](img82.png)
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The unique generalized inverse satisfying
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is known as the Moore Penrose pseudoinverse.