Approximation of singularity sets with analytic
graphs over the ball in C2.
Abstract
Theorem 1.
Let B2 be the open unit ball in C2 and
let c(B2) be its closure. Suppose f is a smooth complex
function given in a neighborhood of c(B2) which satisfies
the following property: for every complex line L in
C2 there exists a polynomial PL on L such
that |PL(z)-f(z)| < r for all z in the intersection of L
and c(B2). Suppose in addition that the gradient of f has
modulus <1. Then there exists an analytic polynomial F such
that |F(z)-f(z)|<13 r½ for all z in c(B2).
We use Theorem 1 to prove
Theorem 2. Let f be a
smooth complex function given in a neighborhood of c(B2)
whose gradient has modulus <1 and let K be
a compact set contained in the tube {(z,w) in (c(B2) ×
C) | |w-f(z)| < r} such that K is a singularity set. Then
there exists an analytic polynomial F in C2 with
|F(z)-f(z)| < 26 r½ on the closed ball.