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Using experimentally determined oscillator strengths and photoionization cross-sectional data, we compute the dispersion characteristics of Ar, Kr, and Xe up to their first ionization levels and determine the spectral regions in the VUV where these gases exhibit negative dispersion and so can be efficiently used for frequency tripling. We then investigate the bandwidths over which efficient tripling can be achieved in phase-matched gas mixtures. The bandwidth is limited by the rapidly varying dispersion in the vicinity of resonance transitions in the gases. In particular, we look at the case of frequency tripling 3647 Å radiation to 1215.7 Å (hydrogen Lyman-α) and show, that for fundamental wavelength bandwidths as narrow as 1 Å, the rapid change in refractive index with wavelength can preclude phase matching over the entire bandwidth of the radiation.