  This paper introduces Fourier duality for a class of affine iterated function
systems (IFS) T_i. These systems are determined by a finite family of
contractive affine maps in R^d. Our Fourier duality applies to the resulting
probability measure mu which is fixed by (T_i). When the IFS is given, the
support of the associated mu is a compact set X in R^d, typically a fractal.
Our Fourier duality refers to the Hilbert space L^2(X, mu): We show that under
a certain unitarity condition involving a pair of affine iterated function
systems (T_i) and (S_j) it is possible to recursively construct a Fourier bases
in the Hilbert space L^2(X, mu) with the Fourier basis for one depending on the
other.
