  Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed
function appear in many parts of harmonic analysis, e.g., wavelet analysis and
the analysis of Triebel-Lizorkin spaces. We give a simple characterization of
the functions and matrices yielding such a partition of unity. For invertible
expanding matrices, the characterization leads to easy ways of constructing
appropriate functions with attractive properties like high regularity and small
support. We also discuss a class of integral transforms that map functions
having the partition of unity property to functions with the same property. The
one-dimensional version of the transform allows a direct definition of a class
of nonuniform splines with properties that are parallel to those of the
classical B-splines. The results are illustrated with the construction of dual
pairs of wavelet frames.
