  We introduce a class of functions that limit to multifractal measures and
which arise when one takes the Fourier transform of the Hadamard transform.
This introduces generalizations of the Fourier transform of the well-studied
and ubiquitous Thue-Morse sequence, and introduces also generalizations to
other intriguing sequences. We show their relevance to quantum chaos, by
displaying quantum eigenfunctions of the quantum bakers map that are
approximated well by such measures, thereby extending our recent work where we
pointed to the existence of ``Thue-Morse'' states.
