TITLE: elementwise functions of positive definite matrix
QUESTION [11 upvotes]: The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) that if $A$ is PSD, then so is $B=A_f,$ such that $b_{ij} = f(a_{ij}),$ where $f$ is an analytic function all of whose coefficients are positive. The question is, is this result sharp, or are there other $f$ which maps the PSD cone into itself by being element wisely applied?

REPLY [4 votes]: Schoenberg's theorem was first shown without the continuity assumption by Rudin in this paper in Duke Math. J., in 1959. In particular, prior to the work of Herz cited above.