Abstract :

In complex analysis, quadrature domains are those domains for which the integral of every function in the Bergman class of functions can be written as a fixed quadrature identity. In this talk, we prove a density theorem for quadrature domains in $\mathbb{C}^n, n \geq 2$. It is shown that quadrature domains are dense in the class of all product domains of the form $D\times \Omega$ where $D \subset\mathbb{C}^{n-1}$ is a smoothly bounded pseudo-convex domain satisfying Bell's Condition R and $\Omega \subset \mathbb{C}$ is a smoothly bounded domain. We shall also discuss a recent result disproving a conjecture by Steven Bell.