Nov 27, 2013.

We consider some observation operator $H: X\rightarrow Y$ defined on the space $X$. Our goal is to solve an operator equation of the type \begin{equation} H \varphi = f \end{equation} with $f \in Y$ given.

Our question is: can we find some transformation $T: X \rightarrow X$, such that $\tilde{H} := H T^{-1}$ is local?

By singular value decomposition SVD we have $$ H = U \Lambda V^{T}, $$ where $\Lambda$ is a digonal matrix and the matrices $U$ and $V$ consist of orthonormal vectors. Now, we define $$ T := V^{T} $$ and $$ \tilde{f} = U \overline f

$$ \tilde{\varphi} = T\varphi$$

$$ H\varphi = HT^{-1}T\varphi $$

$$ = H^{\sim}T\varphi = H^{\sim}\varphi^{\sim} $$

such that $H^{\sim}$ is local