  The problem of constructing physically and mathematically well-defined Wigner
functions for the canonical pair angle and angular momentum is solved. While a
key element for the construction of Wigner functions for the planar phase space
(q,p) in R^2 is the Heisenberg-Weyl group, the corresponding group for the
cylindrical phase space (theta,p) in S^1 x R is the Euclidean group E(2) of the
plane and its unitary representations. Here the angle theta is replaced by the
pair (cos theta, sin theta) which corresponds uniquely to the points on the
unit circle. The main structural properties of the Wigner functions for the
planar and the cylindrical phase spaces are strikingly similar. A crucial role
plays the sinc function which provides the interpolation for the discontinuous
quantized angular momenta in terms of the continuous classical ones, in
accordance with the famous Whittaker cardinal function, well-known from
interpolation and sampling theories. The quantum mechanical marginal
distributions for angle (continuous) and angular momentum (discontinuous) are
as usual uniquely obtained by appropriate integrations of the (theta,p) Wigner
function. Among the examples discussed is an elementary system of simple "cat"
states.
