In this final chapter, we present applications of the theory of induced representations where knowing an explicit expression for an induced representation of a specific group is used.

In Section 7.1, we apply Mackey's theory, and one realization of the induced representations giving the irreducible representations to study the asymptotic behavior of coefficient functions of those representations. The main theorem is that those coefficient functions of infinite-dimensional irreducible representations of motion groups vanish at infinity. As a consequence, one can conclude that the image of a motion group under any irreducible representation is closed in the unitary group with the weak operator topology.

Section 7.2 is concerned with introducing methods for constructing self-adjoint idempotents, or projections, in L1(G) for certain kinds of groups G. A key observation is that the support in Ĝ of a projection must be a compact open set. After reviewing how projections arise for compact and abelian G, we turn to the noncompact, nonabelian situation. Drawing upon the theory developed in Chapters 4 and 5, we identify groups with nontrivial compact open sets in their duals. For appropriate groups G and explicit induced representations of those groups, coefficient functions of those representations can be modified to produce nontrivial projections in L1(G).

Finally, certain identities arising in the construction of projections can be exploited to produce generalizations of the continuous wavelet transform. This is explored in Section 7.3.