  This work analyzes the forward and inverse scattering series for scalar waves
based on the Helmholtz equation and the diffuse waves from the time-independent
diffusion equation, which are important PDEs in various applications. Different
from previous works, which study the radius of convergence for the forward and
inverse scattering series, the stability, and the approximation error of the
series under the $L^p$ norms, we study these quantities under the Sobolev $H^s$
norm, which associates with a general class of $L^2$-based function spaces. The
$H^s$ norm has a natural spectral bias based on its definition in the Fourier
domain: the case $s<0$ biases towards the lower frequencies, while the case
$s>0$ biases towards the higher frequencies. We compare the stability estimates
using different $H^s$ norms for both the parameter and data domains and provide
a theoretical justification for the frequency weighting techniques in practical
inversion procedures. We also provide numerical inversion examples to
demonstrate the differences in the inverse scattering radius of convergence
under different metric spaces.
