Pick interpolation in several variables

Hamilton, Ryan

doi:10.1090/s0002-9939-2013-11571-6

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…There have thus been several articles in the last two decades that have dwelt on the problem ( * ). Apart from the aforementioned works, we refer the reader to the articles [9], [12] and to the works listed in the references therein. However, despite all the results obtained so far:…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The three-point Pick–Nevanlinna interpolation problem on the polydisc

Chandel

2017

Complex Variables and Elliptic Equations

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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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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The three-point Pick–Nevanlinna interpolation problem on the polydisc

Chandel

2017

Complex Variables and Elliptic Equations

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“…That approach led to Agler's solution of ( * ) for n = 2: see [1] (see also the articles [5] by Ball-Trent and [2] by Agler-McCarthy). There have been a number of articles, based on largely functional-analytic ideas, in the last two decades that have dwelt on the problem ( * ): we refer the reader to the works listed in the bibliography of [11]. The latter work, we must mention, addresses -using a result of Bercovici-Westwood [6] the problem of characterizing the existence of interpolants in an arbitrary unital weak * closed subalgebra of H ∞ (D n ).…”

Section: N If and Only If The Matricesmentioning

confidence: 99%

Pick Interpolation on the Polydisc: Small Families of Sufficient Kernels

Bharali

Chandel

2017

Complex Anal. Oper. Theory

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Abstract. We give a solution to Pick's interpolation problem on the unit polydisc in C n , n ≥ 2, by characterizing all interpolation data that admit a D-valued interpolant, in terms of a family of positive-definite kernels parametrized by a class of polynomials. This uses a duality approach that has been associated with Pick interpolation, together with some approximation theory. Furthermore, we use duality methods to understand the set of points on the n-torus at which the boundary values of a given solution to an extremal interpolation problem are not unimodular.

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“…They even failed for higher dimensional polydiscs D n . Some results for D n and the Euclidean ball B n were obtained by Hamilton [13]. Interpolation in the Euclidean ball was also investigated by Amar and Thomas [7,8].…”

Section: Introductionmentioning

confidence: 99%