  This survey paper examines the work of J. von Neumann and M.H. Stone as it
relates to the abstract theory of wavelets. In particular, we discuss the
direct integral theory of von Neumann and how it can be applied to
representations of certain discrete groups to study the existence of normalized
tight frames in the setting of Gabor systems and wavelets, via the use of group
representations and von Neumann algebras. Then the extension of Stone's theorem
due to M. Naimark, W. Ambrose and R. Godement is reviewed, and its relationship
to the multiresolution analyses of S. Mallat and Y. Meyer and the generalized
multiresolution analyses of L. Baggett, H. Medina, and K. Merrill. Finally, the
paper ends by discussing some recent work due to the author, Baggett, P.
Jorgensen and Merrill, and its relationship to operator theory.
