WAVELET ANALYSIS OF INHOMOGENEOUS DATA WITH APPLICATION TO THE COSMICVELOCITY-FIELD

In this paper we give an account of a method of smoothing spatial inho mogeneous data sets by using wavelet reconstruction on a regular grid in an auxilliary space onto which the original data is mapped. In a pr evious paper by the present authors, we devised a method for inferring the velocity potential from the radial component of the cosmic veloci ty field assuming an ideal sampling. Unfortunately the sparseness of t he real data (the peculiar velocities of galaxies) as well as errors o f measurement require us to first smooth the velocity field as observe d on a three-dimensional support (i.e. the galaxy positions) inhomogen eously distributed throughout the sampled volume. The wavelet formalis m permits us to introduce a minimal smoothing procedure that is charac terized by the variation in size of the smoothing window function. Mor eover, the output-smoothed radial velocity field can be shown to corre spond to a well defined theoretical quantity as long as the spatial sa mpling support satisfies certain criteria. We also argue that one shou ld be very cautious when comparing the velocity potential derived from such a smoothed radial component of the velocity field with related q uantities derived from other studies (e.g. of the density field).