  Certain Bernoulli convolution measures (\mu) are known to be spectral.
Recently, much work has concentrated on determining conditions under which
orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure
known to be spectral, the orthonormal basis need not be unique; indeed, there
are often families of such spectral bases.
  Let \lambda = 1/(2n) for a natural number n and consider the Bernoulli
measure (\mu) with scale factor \lambda. It is known that L^2(\mu) has a
Fourier basis. We first show that there are Cuntz operators acting on this
Hilbert space which create an orthogonal decomposition, thereby offering
powerful algorithms for computations for Fourier expansions.
  When L^2(\mu) has more than one Fourier basis, there are natural unitary
operators U, indexed by a subset of odd scaling factors p; each U is defined by
mapping one ONB to another. We show that the unitary operator U can also be
orthogonally decomposed according to the Cuntz relations. Moreover, this
operator-fractal U exhibits its own self-similarity.
