  We present a general method of constructing Winf and winf gauge theories in
terms of d+2 dimensional local fields. In this formulation the \Winf gauge
theory Lagrangians involve non-local interactions, but the winf theories are
entirely local. We discuss the so-called classical contraction procedure by
which we derive the Lagrangian of winf gauge theory from that of the
corresponding Winf gauge theory. In order to discuss the relationship between
quantum Winf and quantum winf gauge theory we solve d=1 gauge theory models of
a Higgs field exactly by using the collective field method. Based on this we
conclude that the Winf gauge theory can be regarded as the large N limit of the
corresponding SU(N) gauge theory once an appropriate coupling constant
renormalization is made, while the winf gauge theory cannot be.
