Fatou theorem on nontangential limits and questions of extension on the ideal boundary

The positive answer is given to a question of S. Axler and A. Shields: it is possible to extend continuously on the Shilov boundary $M(L^\infty)$ of the algebra $H^\infty$ arbitrary continuous boundary function in the unit disc having almost everywhere non-tangential limits. So, a description is obtained, in terms of continuous extension on the part of boundary $M(H^\infty)$; of the maximum class of continuous functions satisfying the Fatou theorem on non-tangential limits.