  We present a new proof of a theorem of Mallat which describes a construction
of wavelets starting from a quadrature mirror filter. Our main innovation is to
show how the scaling function associated to the filter can be used to identify
a certain direct limit of Hilbert spaces with $L^2(\R)$ in such a way that one
can immediately identify the wavelet basis. Our arguments also use a pair of
isometries introduced by Bratteli and Jorgensen, and exploit the geometry
inherent in the Cuntz relations satisfied by these isometries.
