  Schoenberg showed that a function $f:(-1,1)\rightarrow \mathbb{R}$ such that
$C=[c_{ij}]_{i,j}$ positive semi-definite implies that $f(C)=[f(c_{ij})]_{i,j}$
is also positive semi-definite must be analytic and have Taylor series
coefficients nonnegative at the origin. The Schoenberg theorem is essentially a
theorem about the functional calculus arising from the Schur product, the
entrywise product of matrices. Two important properties of the Schur product
are that the product of two rank one matrices is rank one, and the product of
two positive semi-definite matrices is positive semi-definite. We classify all
products which satisfy these two properties and show that these generalized
Schur products satisfy a Schoenberg type theorem.
