  The goal of harmonic analysis on a (noncommutative) group is to decompose the
most `natural' unitary representations of this group (like the regular
representation) on irreducible ones. The infinite-dimensional unitary group
U(infinity) is one of the basic examples of `big' groups whose irreducible
representations depend on infinitely many parameters. Our aim is to explain
what the harmonic analysis on U(infinity) consists of.
  We deal with unitary representations of a reasonable class, which are in 1-1
correspondence with characters (central, positive definite, normalized
functions on U(infinity)). The decomposition of any representation of this
class is described by a probability measure (called spectral measure) on the
space of indecomposable characters. The indecomposable characters were found by
Dan Voiculescu in 1976.
  The main result of the present paper consists in explicitly constructing a
4-parameter family of `natural' representations and computing their characters.
We view these representations as a substitute of the nonexisting regular
representation of U(infinity). We state the problem of harmonic analysis on
U(infinity) as the problem of computing the spectral measures for these
`natural' representations. A solution to this problem is given in the next
paper math/0109194, joint with Alexei Borodin.
  We also prove a few auxiliary general results. In particular, it is proved
that the spectral measure of any character of U(infinity) can be approximated
by a sequence of (discrete) spectral measures for the restrictions of the
character to the compact unitary groups U(N). This fact is a starting point for
computing spectral measures.
