Galaxies and Supermassive Black Holes at z <= 0.1: The Velocity
  Dispersion Function
  We study the distribution of central velocity dispersion, sigma, for >100000
galaxies in the SDSS at 0.01 <= z <= 0.1. We construct the velocity dispersion
function (VDF) from samples complete for all sigma, where galaxies with sigma
greater than the sigma-completeness limit of the SDSS spectroscopic survey are
included. We compare two different sigma estimates; one based on SDSS
spectroscopy (sigma_spec) and another on photometric estimates (sigma_mod). The
sigma_spec for our sample is systematically higher than sigma_mod for all
ranges of sigma, implying that rotational velocity may affect sigma_spec
measurements. The VDFs measured from these quantities are remarkably similar
for lower sigma values, but the sigma_mod VDF falls faster than the sigma_spec
VDF at log sigma > 2.35. Very few galaxies are observed to have sigma > 350
km/s. Despite differences in sample selection and methods, our VDFs are in
close agreement with previous determinations for the local universe, and our
results confirm that complete sampling is necessary to accurately discern the
shape of the VDF at all ranges. We also find that both late and early type
galaxies have sigma_spec > sigma_mod, suggesting that the rotation component of
most galaxies figure significantly into sigma_spec measurements. Early-type
galaxies dominate the population of high sigma galaxies, while late-type
galaxies dominate the low sigma statistic. Our results warrant a more thorough
and cautious approach in using long-slit spectroscopy to derive the statistics
of local galaxies. Higher quality photometric measurements will enable more
accurate and less uncertain measurements of the sigma_mod VDF, as described
here. A follow-up paper uses the final samples from this work in conjunction
with the MBH-sigma relation to derive the z<=0.1 black hole mass function
(BHMF).
