Approximation by Ratios of Bounded Analytic Functions

Let D be the unit disk andEbe a compact subset of theH∞maximal ideal space,M(H∞). We show that any continuous functionfon an open neighborhoodU⊂M(H∞) ofEsuch thatf |U∩D∈H∞(U∩D) can be uniformly approximated onEby ratiosh/g, whereh, g∈H∞andgis zero free onE. We also characterize those setsEfor whichh/gcan be replaced byhfor everyf. Finally, we prove that any inner functionucan be written as a rational function of interpolating Blaschke products. It follows thatucan be uniformly approximated by ratios of interpolating Blaschke products on any compact setE⊂M(H∞) whereuis zero free.