publication:

Test Functions, Kernels, Realizations and Interpolation (2007)

Author(s): Dritschel MA, McCullough S

    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.

      • Book Title: Operator Theory, Structured Matrices, and Dilations: Tiberiu Constantinescu Memorial Volume
      • Publisher: American Mathematical Society
      • Publication type: Book chapter
      • Bibliographic status: Published

        Keywords: Agler-Schur class, interpolation, Nevanlinna-Pick, transfer functions, test functions, multivariable, multiply connected domains, Stone-\v{C}ech compactification

        Staff

        Dr Michael Dritschel
        Reader in Pure Mathematics