Operator theory and the Oka extension theorem

For $\delta$ an $m$-tuple of analytic functions, we define an algebra $\text{hidg}$, contained in the bounded analytic functions on the analytic polyhedron $\mathrm{\mid }{\delta }^l(z)\mathrm{\mid }<1,1\le l\le m$, and prove a representation formula for it. We give conditions whereby every function that is analytic on a neighborhood of $\mathrm{\mid }{\delta }^l(z)\mathrm{\mid }\le 1,1\le l\le m$ is actually in $\text{hidg}$. We use this to give a proof of the Oka extension theorem with bounds. We define an $\text{hidg}$ functional calculus for operators.