Interpolation by Bounded Analytic Functions

We show that Fatou's interpolation theorem of 1906 directly implies: (i) the Rudin-Carleson interpolation theorem, (ii) a theorem of Lohwater and Piranian on the radial limits of bounded analytic functions; and (iii) a known approximation lemma by R. Doss. We also present some new interpolation results in the unit disc, when the interpolation of a continuous function on a closed set on the unit circle is required to be done by the limiting values of a bounded analytic function.