  For some fractal measures it is a very difficult problem in general to prove
the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In
fact there are examples of extremely sparse sets that are not even Bessel
spectra. In this paper we investigate this problem for general fractal measures
induced by iterated function systems (IFS). We prove some existence results of
spectra associated with Hadamard pairs. We also obtain some characterizations
of Bessel spectrum in terms of finite matrices for affine IFS measures, and one
sufficient condition of frame spectrum in the case that the affine IFS has no
overlap.
