The Frostman Shift of an inner function at the value which assumed infinitely and is not an asymptomatic value is an infinite Blaschke Product. But how to charaterize it when it is an asymptotic value?
By Fatou's theorem, the radial limit function $$\phi^{\ast}(\zeta):=\lim_{r\rightarrow1^{-}}\phi(r\zeta),$$ for a bounded analytic function $\phi$ on $\mathbb{D}$, exists for $m$-almost every $\zeta\in\partial\mathbb{D}$(here $m$ is a normalized Lebesgue measure on $\partial\mathbb{D}$). If $|\phi^{\ast}(\zeta)|=1$ for almost every $\zeta$, then $\phi$ is called an inner function.
An inner function can be factored as $$\phi(z)=e^{i\gamma}z^pB(z)\exp(-\int_{\partial\mathbb{D}}\frac{\zeta+z}{\zeta-z}d\mu(\zeta)).$$ Here $\mu$ is a positive finite measure on $\partial\mathbb{D}$ with $\mu\bot m$ and $B(z)$ is a Blaschke Product.
The Frostman shift $$\phi_a(z):=\tau_a\circ\phi(z)=\frac{\phi(z)-a}{1-\overline{a}\phi(z)}, |a|<1$$ are certainly inner functions when $\phi(z)$ is an inner function.