The classical theorems of Mittag-Leffler and Weierstrass show that when \((\lambda _n)_{n \geqslant 1}\) is a sequence of distinct points in the open unit disk \(\mathbb {D}\), with no accumulation points in \(\mathbb {D}\), and \((w_n)_{n \geqslant 1}\) is any sequence of complex numbers, there is an analytic function \(\varphi \) on \(\mathbb {D}\) for which \(\varphi (\lambda _n) = w_n\). A celebrated theorem of Carleson [2] characterizes when, for a bounded sequence \((w_n)_{n \geqslant 1}\), this interpolating problem can be solved with a bounded analytic function. A theorem of Earl [5] goes further and shows that when Carleson’s condition is satisfied, the interpolating function \(\varphi \) can be a constant multiple of a Blaschke product. Results from [4] determine when the interpolating function \(\varphi \) can be taken to be zero free. In this paper we explore when \(\varphi \) can be an outer function.

Mittag-Leffler 和 Weierstrass 的经典定理表明，当\((\lambda _n)_{n \geqslant 1}\)是开元圆盘\(\mathbb {D}\)中不同点的序列时，有没有在累积点\（\ mathbb {d} \） ，和\（（w_n）_ {N \ geqslant 1} \）是复数的任何序列，有一个解析函数\（\ varphi \）上\（ \mathbb {D}\)其中\(\varphi (\lambda _n) = w_n\)。一个著名的 Carleson 定理 [2] 描述了对于有界序列\((w_n)_{n \geqslant 1}\)，这个插值问题可以用有界解析函数解决。Earl [5] 的定理更进一步，表明当满足 Carleson 条件时，插值函数\(\varphi \)可以是 Blaschke 乘积的常数倍。[4] 的结果确定了何时可以将插值函数\(\varphi \)视为零自由。在本文中，我们探讨了\(\varphi \)何时可以是外部函数。