We provide a necessary condition for the existence of a 3-point holomorphic interpolant F : D −→ Ωn, n ≥ 2. Our condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma. We combine some of the ideas and techniques used in our result on the O(D, Ωn)-interpolation problem to establish a Schwarz lemma -which may be of independent interest -for holomorphic correspondences from D to a general planar domain Ω ⋐ C.
Given a domain Ω in C m , and a finite set of points z 1 , z 2 , . . . , z n ∈ Ω and w 1 , w 2 , . . . , w n ∈ D (the open unit disc in the complex plane), the Pick interpolation problem asks when there is a holomorphic functionPick gave a condition on the data {z i , w i : 1 ≤ i ≤ n} for such an interpolant to exist if Ω = D. Nevanlinna characterized all possible functions f that interpolate the data. We generalize Nevanlinna's result to a domain Ω in C m admitting holomorphic test functions when the function f comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains -the bidisc, the symmetrized bidisc and the annulus -which are affiliated to given kernels.
Abstract. We give a characterization for the existence of a holomorphic interpolant on the unit polydisc D n , n ≥ 2, for prescribed three-point Pick-Nevanlinna data. One of the key steps is a characterization for the existence of an interpolant that is a rational inner function on D n . The latter reduces the search for a three-point interpolant to finding a single rational inner function that satisfies a type of positivity condition and arises from a polynomial of a very special form. This in turn relies on a pair of results, which are of independent interest, on the factorization of rational inner functions.
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