A smooth (𝒞∞) function on a
smooth real submanifold M of complex Euclidean space Cn is a CR function if it
satisfies the Cauchy-Riemann equations tangential to M. It is shown that each CR
function admits an extension to an open neighborhood of M in Cn whose
z-derivatives all vanish on M to a prescribed high order, provided that the system of
tangential Cauchy-Riemann equations has minimal rank throughout M.
This result is applied to show that on a holomorphically convex compact
set in M each CR fuction can be uniformly approximated by holomorphic
functions.
