  We extend the position-space renormalization procedure, where renormalization
factors are calculated from Green's functions in position space, by introducing
a technique to take the average of Green's functions over spheres. In addition
to reducing discretization errors, this technique enables the resulting
position-space correlators to be evaluated at any physical distance, making
them continuous functions similar to the $O(4)$-symmetric position-space
Green's functions in the continuum theory but with a residual dependence on a
regularization parameter, the lattice spacing $a$. We can then take the
continuum limit of these renormalized quantities calculated at the same
physical renormalization scale $|x|$ and investigate the resulting
$|x|$-dependence to identify the appropriate renormalization window.
  As a numerical test of the spherical averaging technique, we determine the
renormalized light and strange quark masses by renormalizing the scalar
current. We see a substantial reduction of discretization effects on the scalar
current correlator and an enhancement of the renormalization window. The
numerical simulation is carried out with $2+1$-flavor domain-wall fermions at
three lattice cutoffs in the range 1.79--3.15~GeV.
