Approximation on disks

Abstract:

Let $D$ be a closed disk in the complex plane, $f$ a complex valued continuous function on $D$ and ${R_f}(D) =$ the uniform closure on $D$ of rational functions in $z$ and $f$ which are finite. Among other results we obtain the following. Theorem. If $f$ is of class ${C^1}$ in a neighborhood of $D$ and $|{f_{\bar z}}| > |{f_z}|$ everywhere (i.e., $f$ is an orientation reversing immersion of $D$ in the plane), then ${R_f}(D) = C(D)$. Theorem. Let $f$ be a polynomial in $z$ and $\bar z$. If for each a in $D,f - \Sigma {(j!)^{ - 1}}{D^j}f(a){(z - a)^j} = {(\bar z - \bar a)^k}g$ with $|{g_{\bar z}}| > |{g_z}|$ at he zeros of $g$ in $D$ where $Df = {f_z}$, then ${R_f}(D) = C(D)$. Corollary. Let $f$ be a polynomial in $z$ and $\bar z$ and let $|{f_{z\bar z}}(0)| < |{f_{\bar z \bar z}}(0)|/2$. Then there exists an $r > 0$ such that, for $D = (|z| \leqslant r),{R_f}(D) = C(D)$. The proofs of the theorems use measures and the conditions involved in the theorems are independent of each other. Concerning the corollary, results of E. Bishop and G. Stolzenberg show that ${f_{\bar z}}(0) = 0$ and $|{f_{\bar z \bar z}}(0)| < |{f_{z\bar z}}(0)|$, then there exists no $r$ such that ${R_f}(D) = C(D)$ where $D = (|z| \leqslant r)$. Let $F = ({f_1}, \cdots ,{f_n})$ be a map on $B$ = unit polydisk in ${{\mathbf {C}}^n}$ with values in ${{\mathbf {C}}^n},{P_F}$ = uniform closure on $B$ of polynomials in ${z_1}, \cdots {z_n},{f_1}, \cdots ,{f_n}$. Theorem. If $F$ is of class ${C^1}$ in a neighborhood of $B,{F_{\bar z}}$ is invertible and if for each $a = ({a_1}, \cdots ,{a_n})$ in $B$, there exist complex constants $\{ {c_j}\} ,\{ {d_{ij}}\} ,i,j = 1, \cdots ,n$, such that $\Sigma {c_j}({z_j} - {a_j})({f_j}(z) - {f_j}(a)) + \Sigma {d_{ij}}({z_i} - {a_i})({z_j} - {a_j})$ has positive real part for all $z \ne a$, then $\{ (\zeta ,F(\zeta )):\zeta \in B\}$ is a polynomially convex set. Corollary. If $F = (f,g)$ where $f(z,w) = \bar z + cz\bar z + d{\bar z^2} + q\bar zw,g(z,w) = \bar w + sw\bar w + t{\bar w^2} + p\bar wz$ and the coefficients satisfy $|\bar c + d| + |d| + |q| < 1$ and $|\bar s + t| + |t| + |p| < 1$, then ${P_F} = C(B)$. Corollary. If $F(z) = \bar z + R(z)$ where $R = ({R_1}, \cdots ,{R_n})$ is of class ${C^2}$ and satisfies the Lipschitz condition $|R(\zeta ) - R(\eta )| \leqslant k|\zeta - \eta |$ with $k < 1$, then ${P_F} = C(B)$. This last corollary is a result of Hörmander and Wermer. The proof of the theorem uses methods from several complex variables.