Test Functions, Kernels, Realizations and Interpolation

2006/12/20 by Michael A. Dritschel, Scott McCullough, Dritschel, Michael A. +1 · 2 citations
Mathematics · #46E22 (Secondary) #47A13 #47A57 (Primary) #47B38 #47D25 #47L55 #47L75 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.math/0612608

openalex publication_date 2006/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Jim Agler revolutionized the area of Pick interpolation with his realization theorem for what is now called the Agler-Schur class for the unit ball in \mathbb C^d. We discuss an extension of these results to algebras of functions arising from test functions and the dual notion of a family of reproducing kernels, as well as the related interpolation theorem. When working with test functions, one ideally wants to use as small a collection as possible. Nevertheless, in some situations infinite sets of test functions are unavoidable. When this is the case, certain topological considerations come to the fore. We illustrate this with examples, including the multiplier algebra of an annulus and the infinite polydisk.

Cited by

• Schur-Agler class and Carathéodory extremal functions
• Function Theory on Tetrablock: Realization, Interpolation, Extension and Toeplitz Corona Theorem

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