An Approximation Theorem

Both the analytic disc and Fourier transform approaches to CR extension currently use an approximation theorem by Baouendi and Treves. This theorem roughly states that CR functions on a submanifold of ℂ ⁿ can be locally approximated by entire functions on ℂ ⁿ . This approximation theorem has an interesting history. There are earlier versions of this theorem that are weaker (and some with errors in their proofs). All of these earlier versions have some convexity assumptions on the Levi form. Then around 1980, Baouendi and Treves caught the CR extension community by surprise. They showed that no convexity assumptions are needed. Moreover, their proof is a simple but clever adaptation of the proof of the classical Weierstrass theorem on approximating continuous functions on ℝ ⁿ by polynomials (see Section 1.1).