This question follows from another question I asked here.
I am currently trying to read this paper and I am having difficulty in understanding the interpretations of eq. 7 which are given in lines immediately after the equation. It would be very helpful if you can provide hints on what they mean and also the calculation of the $v_i$ and $\lambda_i$.
As I understand, the point that the author is trying to make is that eigenvectors are localized in either the partial sky or the sky which is masked. But I am unable to follow how the inference is derived.
Any help in this regards would be appreciated.
Update 1 Jan 2021 So, here is the math that I did.The integral in equation 7 of the paper that I linked to goes like this, $$ \int_{S'} v_i(\pmb{\hat{r}}')\delta(\pmb{\hat{r}}-\pmb{\hat{r}}')d\Omega' = \lambda_iv_i(\pmb{\hat{r}})$$ So, the unit vector $\pmb{\hat{r}}$ can point to any direction (in the partial sky $S'$ covered or in the cut sky $S-S'$. The paragraph immediately after this equation mentions that for a given eigenvector, this equation is true for any direction $\pmb{\hat{r}}$. Now, if we take that the $\pmb{\hat{r}}$ is in the partial sky $S'$, then the integral (due to the dirac delta function would just be 1 at the location when $\pmb{\hat{r}}$=$\pmb{\hat{r}}'$ and zero at all other directions which just reduces the equation to, $$v_i(\pmb{\hat{r}}) = \lambda_iv_i(\pmb{\hat{r}})$$, which implies that $v_i(\pmb{\hat{r}}) \neq 0$ but $\lambda_i = 1$. On the other hand, when $\pmb{\hat{r}}$ \neq $\pmb{\hat{r}}'$ then the dirac delta on the left hand side evaluates to zero, hence $$\lambda_iv_i(\pmb{\hat{r}}) = 0$$ In the first case, it is rather easy to see that $\lambda_i$ is 1. But, I am unable to understand all the other results and constraints of $v_i(\pmb{\hat{r}})$. I cant use $v_i(\pmb{\hat{r}})$ for the two directions together because they are in principle different directions. So, how do I get the result of the paper that, for cut region ($S-S'$) $v_i(\pmb{\hat{r}}) = 0$ and $\lambda_i = 1$ and in the partial covered region, $S'$, $v_i(\pmb{\hat{r}}) = 0$ and $\lambda_i = 0$.
Due to the Dirac delta, the evaluation of the integral separates into the two cases, depending on whether $\hat{}$ is in the partial sky or not: $$ \lambda_i v_i(\hat{}) = \int_{S'} v_i(\hat{}') \, \delta(\hat{} - \hat{}') \, \mathrm{d}\Omega' = \begin{cases} v_i(\hat{}) & \text{if $\hat{} \in S'$} \\ 0 & \text{if $\hat{} \notin S'.$} \end{cases} $$ The key, as the paper suggests, is that this equation holds for every $\hat{}$ in the full sky $S$, as $v_i$ is supposed to be an eigenfunction of the operator with eigenvalue $\lambda_i$. The $\hat{} \in S'$ case of the equation, which can be written perhaps more clearly as $(\lambda_i - 1) v_i(\hat{}) = 0$, implies that either
(a) $\lambda_i = 1$
or
(b) $v_i = 0$ for all $\hat{} \in S'$,
while the $\hat{} \notin S'$ case implies either
(c) $\lambda_i = 0$
or
(d) $v_i = 0$ for all $\hat{} \notin S'$.
Since the zero function is not considered an eigenfunction, the only compatible pairs of possibilities are (a)+(d), or (b)+(c), as claimed.