  Primitive inflation tilings of the real line with finitely many tiles of
natural length and a Pisot--Vijayaraghavan unit as inflation factor are
considered. We present an approach to the pure point part of their diffraction
spectrum on the basis of a Fourier matrix cocycle in internal space. This
cocycle leads to a transfer matrix equation and thus to a closed expression of
matrix Riesz product type for the Fourier transforms of the windows for the
covering model sets. In general, these windows are complicated Rauzy fractals
and thus difficult to handle. Equivalently, this approach permits a
construction of the (always continuously representable) eigenfunctions for the
translation dynamical system induced by the inflation rule. We review and
further develop the underlying theory, and illustrate it with the family of
Pisa substitutions, with special emphasis on the Tribonacci case.
