Finite frequency range Kramers Kronig relations: Bounds on the dispersion

Dispersion inequalities are presented to check for the self-consistency of experimentally obtained complex moduli, such as the complex dielectric constant, magnetic permeability, and complex bulk and shear moduli of viscoelastic materials. Unlike the Kramers-Kronig dispersion relations, they only require measurements over a finite frequency range. They can provide highly accurate interpolation formulas for the real part, given its value at a few selected frequencies and given the imaginary part over a range of frequencies.