# Continuity and convergence properties of extremal interpolating disks.

• Volume: 39, Issue: 2, page 335-347
• ISSN: 0214-1493

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## Abstract

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Let a be a sequence of points in the unit ball of Cn. Eric Amar and the author have introduced the nonnegative quantity ρ(a) = infα infk Πj:j≠k dG(αj, αk), where dG is the Gleason distance in the unit disk and the first infimum is taken over all sequences α in the unit disk which map to a by a map from the disk to the ball.The value of ρ(a) is related to whether a is an interpolating sequence with respect to analytic disks passing through it, and if a is an interpolating sequence in the ball, then ρ(a) &gt; 0.In this work, we show that ρ(a) can be obtained as the limit of the same quantity for the truncated finite sequences, and that ρ(a) depends continuously on a when a is finite. Furthermore, we describe some of the behavior of the minimizing sequences of maps involved in the extremal problem used to define ρ.

## How to cite

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Thomas, Pascal J.. "Continuity and convergence properties of extremal interpolating disks.." Publicacions Matemàtiques 39.2 (1995): 335-347. <http://eudml.org/doc/41232>.

@article{Thomas1995,
abstract = {Let a be a sequence of points in the unit ball of Cn. Eric Amar and the author have introduced the nonnegative quantity ρ(a) = infα infk Πj:j≠k dG(αj, αk), where dG is the Gleason distance in the unit disk and the first infimum is taken over all sequences α in the unit disk which map to a by a map from the disk to the ball.The value of ρ(a) is related to whether a is an interpolating sequence with respect to analytic disks passing through it, and if a is an interpolating sequence in the ball, then ρ(a) &gt; 0.In this work, we show that ρ(a) can be obtained as the limit of the same quantity for the truncated finite sequences, and that ρ(a) depends continuously on a when a is finite. Furthermore, we describe some of the behavior of the minimizing sequences of maps involved in the extremal problem used to define ρ.},
author = {Thomas, Pascal J.},
journal = {Publicacions Matemàtiques},
keywords = {Funciones analíticas; Funciones de variación acotada; Bola unidad; Conjuntos de interpolación; Función entera; continuity; convergence; extremal-interpolating disks; sequence of points; unit ball; Gleason distance; unit disk; interpolating sequence; analytic disks; sequences of maps; extremal problem},
language = {eng},
number = {2},
pages = {335-347},
title = {Continuity and convergence properties of extremal interpolating disks.},
url = {http://eudml.org/doc/41232},
volume = {39},
year = {1995},
}

TY - JOUR
AU - Thomas, Pascal J.
TI - Continuity and convergence properties of extremal interpolating disks.
JO - Publicacions Matemàtiques
PY - 1995
VL - 39
IS - 2
SP - 335
EP - 347
AB - Let a be a sequence of points in the unit ball of Cn. Eric Amar and the author have introduced the nonnegative quantity ρ(a) = infα infk Πj:j≠k dG(αj, αk), where dG is the Gleason distance in the unit disk and the first infimum is taken over all sequences α in the unit disk which map to a by a map from the disk to the ball.The value of ρ(a) is related to whether a is an interpolating sequence with respect to analytic disks passing through it, and if a is an interpolating sequence in the ball, then ρ(a) &gt; 0.In this work, we show that ρ(a) can be obtained as the limit of the same quantity for the truncated finite sequences, and that ρ(a) depends continuously on a when a is finite. Furthermore, we describe some of the behavior of the minimizing sequences of maps involved in the extremal problem used to define ρ.
LA - eng
KW - Funciones analíticas; Funciones de variación acotada; Bola unidad; Conjuntos de interpolación; Función entera; continuity; convergence; extremal-interpolating disks; sequence of points; unit ball; Gleason distance; unit disk; interpolating sequence; analytic disks; sequences of maps; extremal problem
UR - http://eudml.org/doc/41232
ER -

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