Unimodular functions and interpolating Blaschke products

The result by Bourgain that every unimodular function $\psi$ on the unit circle can be factored as $\psi = e^{i \tilde v} B_1 \overline B_2$ with $B_1$ and $B_2$ Blaschke products can be improved. We show that the same result holds with $B_1$ and $B_2$ interpolating Blaschke products. This will at the same time be a refinement of Jones's result that every unimodular function can be approximated in the $H^\infty$-norm by a ratio of interpolating Blaschke products.