  We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl
transform and prove three instances thereof, one of which involves a limit
transition from Opdam's results for the graded Hecke algebra. Furthermore, the
connection between Dunkl operators and the Cartan motion group is established.
It is shown how the algebra of radial parts of invariant differential operators
can be described explicitly in terms of Dunkl operators, which implies that the
generalized Bessel functions coincide with the spherical functions. In this
context, the restriction of Dunkl's intertwining operator to the invariants can
be interpreted in terms of the Abel transform. Using shift operators we also
show that, for certain values of the multiplicities of the restricted roots,
the Abel transform is essentially inverted by a differential operator.
