Wavelet analysis of inhomogeneous data with application to the cosmic velocity field

Abstract

An account is given of a method of smoothing spatial inhomogeneous data sets by using wavelet reconstruction on a regular grid in an auxilliary space onto which the original data is mapped. In a previous paper by the present authors, the authors devised a method for inferring the velocity potential from the radial component of the cosmic velocity field assuming an ideal sampling. Unfortunately the sparseness of the real data (the peculiar velocities of galaxies) as well as errors of measurement requires that the velocity field is first smoothed as observed on a three-dimensional support (i.e. the galaxy positions) inhomogeneously distributed throughout the sampled volume. The wavelet formalism permits a minimal smoothing procedure to be introduced that is characterized by the variation in size of the smoothing window function. Moreover, the output-smoothed radial velocity field can be shown to correspond to a well defined theoretical quantity as long as the spatial sampling support satisfies certain criteria. The authors also argue that one should be very cautious when comparing the velocity potential derived from such a smoothed radial component of the velocity field with related quantities derived from other studies (e.g. of the density field).