Vol. 33, No. 1, 1970

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Tangential Cauchy-Riemann equations and uniform approximation

Michael Benton Freeman

Vol. 33 (1970), No. 1, 101–108

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.

Mathematical Subject Classification
Primary: 32.70
Received: 21 November 1969
Published: 1 April 1970
Michael Benton Freeman