We describe a new, multiply subtractive Kramers-Kronig (MSKK)method to find the optical constants of a material from a singletransmittance or reflectance spectrum covering a small frequencydomain. The MSKK method incorporates independent measurements ofn and k at one or more reference wave-numbervalues to minimize errors due to extrapolations of the data. Anunexpected connection between the MSKK equations and the interpolationtheory allows us to derive the equations from an interpolationtheorem. We found that the locations of the reference points affectthe accuracy of the values determined for the optical constants andthat the optimal spacing of N reference data points isrelated to the zeros of a suitably transformed Chebychev polynomial oforder N. We discuss our efforts to optimize both the numberand the spacing of these reference points and apply our method to sometest spectra.