  We apply a special case, the restriction principle (for which we give a
definition simpler than the usual one), of a basic result in functional
analysis (the polar decomposition of an operator) in order to define $C_{\mu,
t}$, the $C$-version of the Segal-Bargmann transform, associated to a finite
Coxeter group acting in $\mathbb{R}^N$ and a given value $t>0$ of Planck's
constant, where $\mu$ is a multiplicity function on the roots defining the
Coxeter group. Then we immediately prove that $C_{\mu, t}$ is a unitary
isomorphism. To accomplish this we identify the reproducing kernel function of
the appropriate Hilbert space of holomorphic functions. As consequences we
prove that the Segal-Bargmann transforms for Versions $A$, $B$ and $D$ are also
unitary isomorphisms, though not by a direct application of the restriction
principle. The point is that the $C$-version is the the only version where a
restriction principle, in our definition of this method, applies directly. This
reinforces the idea that the $C$-version is the most fundamental, most natural
version of the Segal-Bargmann transform.
