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.



How to characterize the value which assumed by an inner function infinitely often but is an asymptotic value？
If $a$ is an asymptotic value, the inner function $\phi\_a$ may be a Blaschke product or not. It is not clear what do you mean by "how to chracterize it". In what terms?
It is an inner function
for which zero is an asymptotic value. (This characterizes it. Is this what you want?).


In a special case when all zeros of $\phi\_a$ are real there is a complete answer
(on the question, whether $\phi\_a$ is a Blaschke product or not) in terms of
critical values of $\phi\_a$ (or $\phi$, does not matter). This answer is highly non-trivial, and its generalization to non-real zeros is not known. The characterization
I am talking about is contained in the paper A. A. Goldberg, Some asymptotic properties
of meromorphic functions, Complex variables, 37 (1998) 225--241.
English translation from the Russian original published in 
Uchenye Zapiski Lvovskogo Gosudarstvennogo Universiteta, 38, 7, 
1956. 54-74. If your library does not have these journals, I can send you
a reprint of the translation. If you need it, send an e-mail to me.
(Complex Variables journal is not freely available on the web).


