Two subdivision schemes with Hermite data on Z are studied These schemes us
e 2 or 7 parameters respectively depending on whether Hermite data involve
only first derivatives or include second derivatives, For a large region in
the parameter space, the schemes are convergent in the space of Schwartz d
istributions. The Fourier transform of any interpolating function can be co
mputed through products of matrices of order 2 or 3. The Fourier transform
is related to a specific system of functional equations whose analytic solu
tion is unique except for a multiplicative constant. The main arguments for
these results come from Paley-Wiener-Schwartz theorem on the characterizat
ion of the Fourier transforms of distributions with compact support and a t
heorem of Artzrouni about convergent products of matrices.