  We propose to study deformation quantizations of Whitney functions. To this
end, we extend the notion of a deformation quantization to algebras of Whitney
functions over a singular set, and show the existence of a deformation
quantization of Whitney functions over a closed subset of a symplectic
manifold. Under the assumption that the underlying symplectic manifold is
analytic and the singular subset subanalytic, we determine that the Hochschild
and cyclic homology of the deformed algebra of Whitney functions over the
subanalytic subset coincide with the Whitney--de Rham cohomology. Finally, we
note how an algebraic index theorem for Whitney functions can be derived.
