With overlapping radial window functions (#118, #119), there is no longer a unique way to distribute galaxies among each shell to recover some given galaxy distribution $dN/dz$.
If $w_i$ are the radial window functions, $f = dN/dz$ is the target function, and $z_i$ is a redshift grid, we want a solution $x_i$ of the following overdetermined matrix equation:
With overlapping radial window functions (#118, #119), there is no longer a unique way to distribute galaxies among each shell to recover some given galaxy distribution $dN/dz$.
If $w_i$ are the radial window functions, $f = dN/dz$ is the target function, and $z_i$ is a redshift grid, we want a solution $x_i$ of the following overdetermined matrix equation:
Linear least squares is one obvious method. A weighting $\Delta z_i$ can ensure that finer-sampled regions do not overly influence the result:
The $\Delta z_i$ could be the weights of the trapezoidal rule.