CARLEMAN APPROXIMATION BY ENTIRE-FUNCTIONS ON THE UNION OF 2 TOTALLY-REAL SUBSPACES OF C(N)

Let L1, L2 subset-of C(n) be two totally real subspaces of real dimension n, and such that L1 and L2 = {0}. We show that continuous functions on L1 or L2 allow Carleman approximately by entire functions if and only if L1 or L2 is polynomially convex. If the latter condition is satisfied, then a function f:L1 or L2 --> C such that f\L(i) is-an-element-of C(k)(L(i)), i = 1, 2, allows Carleman approximation of order k by entire functions if and only if f satisfies the Cauchy-Riemann equations up to order k at the origin.