Abstract
Let Θ be an inner function with finitely many singularities on the unit circle. Then, given distinct points α and β of unit modulus, an explicit expression for Θ is given, in terms of the sets of points on the circle at which Θ takes the values α and β. If Θ has just one singularity, then it is unique to within a hyperbolic automorphism: this property need not hold in general. At the same time, a full characterization of the possible sets at which such values may be taken is obtained. The methods used involve an extension of the Hermite-Biehler Theorem to functions with finitely many essential singularities. Further applications in the theory of isometric composition operators, as well as uniqueness results for restricted shifts involving their numerical ranges are discussed.
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