Abstract
Given n, N ≥ 1 we construct a set of points \({\lambda_1,{\ldots},\lambda_{N^n}\in{\mathbb D}^n}\) such that for each rational inner function f on \({{\mathbb D}^n}\) of degree less than N the Pick problem on \({{\mathbb D}^n}\) with data \({\lambda_1,{\ldots},\lambda_{N^n}}\) and \({f(\lambda_1),{\ldots},f(\lambda_{N^n})}\) has a unique solution. In particular, we construct a 1-dimensional inner variety V and show that the points \({\lambda_1,{\ldots},\lambda_{N^n}}\) may be chosen almost arbitrarily in \({V\cap{\mathbb D}^n}\). Our results state that f is uniquely determined in the Schur class of \({{\mathbb D}^n}\) by its values on \({\lambda_1,{\ldots},\lambda_{N^n}}\).
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Communicated by Joseph Ball.
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Scheinker, D. A Uniqueness Theorem for Bounded Analytic Functions on the Polydisc. Complex Anal. Oper. Theory 7, 1429–1436 (2013). https://doi.org/10.1007/s11785-011-0179-5
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DOI: https://doi.org/10.1007/s11785-011-0179-5