The classical realization theorem gives various characterizations of functions in the unit ball of $H^\\infty(\\mathbb D)$, the bounded analytic functions on the unit disk $\\mathbb D$, which happens to also correspond to the multipliers of Hardy space $H^2(\\mathbb D)$. This realization theorem yields an elegant way of solving the Nevanlinna-Pick interpolation problem. In the mid 80s, Jim Agler discovered an analogue of the realization theorem over the polydisk. While for $d=2$, it once again gives a characterization of the elements of the unit ball of $H^\\infty(\\mathbb D^d)$ (and so allows one to solve interpolation problems in $H^\\infty(\\mathbb D^2)$), for $d>2$, the class of functions which are realized is a proper subset of the unit ball of $H^\\infty(\\mathbb D^d)$ — the so-called Schur-Agler class.

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Last updated: 27 Feb 2024