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Abstract 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 partial differential equations (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 theL^pnorms, we study these quantities under the SobolevH^snorm, which associates with a general class ofL²-based function spaces. TheH^snorm has a natural spectral bias based on its definition in the Fourier domain: the cases < 0 biases towards the lower frequencies, while the cases > 0 biases towards the higher frequencies. We compare the stability estimates using differentH^snorms 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.