Approximation theorems on mapping properties of the classical kernel functions of complex analysis

Let $\Omega$ be a bounded domain in $\doubc$ with $C\sp\infty$ smooth boundary and let $w\sb0 \in b\Omega$. We prove that for each $z\sb0 \in b\Omega - \{w\sb0\}$ and each $i \ge 0$, span $\{\partial\sbsp{z}{i}\partial\sbsp{\bar w}{k}K$ ($\cdot,w\sb0) : k \ge 0\}$ is dense in $H\sp\infty(\overline\Omega\sb0$) for any smoothly bounded simply-connected domain $\Omega\sb0 \subset \Omega$ with $z\sb0 \in b\Omega\sb0$ and $w\sb0 \not\in b\Omega\sb0$. We show the similar result for the Szego kernel. Let $\Omega\sb1$ and $\Omega\sb2$ be bounded domains in $\doubc$ with $C\sp\infty$ smooth boundaries and $\Omega\sb2$ be simply-connected. We obtain a simple transformation formula for the Szego kernels under a proper holomorphic mapping of $\Omega\sb1$ to $\Omega\sb2$. We get a transformation formula for the Bergman kernels under a proper anti-holomorphic correspondence between two bounded domains in $\doubc\sp{n}$. We find a formula explaining the relation of the Bergman kernel and the Szego kernel when the given domain is smoothly bounded in $\doubc$.