  The classical Haar construction of Brownian motion uses a binary tree of
triangular wedge-shaped functions. This basis has compactness properties which
make it especially suited for certain classes of numerical algorithms. We
present a similar basis for the Ornstein-Uhlenbeck process, in which the basis
elements approach asymptotically the Haar functions as the index increases, and
preserve the following properties of the Haar basis: all basis elements have
compact support on an open interval with dyadic rational endpoints; these
intervals are nested and become smaller for larger indices of the basis
element, and for any dyadic rational, only a finite number of basis elements is
nonzero at that number. Thus the expansion in our basis, when evaluated at a
dyadic rational, terminates in a finite number of steps. We prove the
covariance formulae for our expansion and discuss its statistical
interpretation and connections to asymptotic scale invariance.
