Extreme and exposed representing measures of the disk algebra

Alex Heinis; Jan Wiegerinck

Annales Polonici Mathematici (2000)

  • Volume: 73, Issue: 2, page 105-118
  • ISSN: 0066-2216

Abstract

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We study the extreme and exposed points of the convex set consisting of representing measures of the disk algebra, supported in the closed unit disk. A boundary point of this set is shown to be extreme (and even exposed) if its support inside the open unit disk consists of two points that do not lie on the same radius of the disk. If its support inside the unit disk consists of 3 or more points, it is very seldom an extreme point. We also give a necessary condition for extreme points to be exposed and show that so-called BSZ-measures are never exposed.

How to cite

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Heinis, Alex, and Wiegerinck, Jan. "Extreme and exposed representing measures of the disk algebra." Annales Polonici Mathematici 73.2 (2000): 105-118. <http://eudml.org/doc/262663>.

@article{Heinis2000,
abstract = {We study the extreme and exposed points of the convex set consisting of representing measures of the disk algebra, supported in the closed unit disk. A boundary point of this set is shown to be extreme (and even exposed) if its support inside the open unit disk consists of two points that do not lie on the same radius of the disk. If its support inside the unit disk consists of 3 or more points, it is very seldom an extreme point. We also give a necessary condition for extreme points to be exposed and show that so-called BSZ-measures are never exposed.},
author = {Heinis, Alex, Wiegerinck, Jan},
journal = {Annales Polonici Mathematici},
keywords = {exposed point; representing measure; extreme point; disk algebra; exposed points; convex set; representing measures; BSZ-measures},
language = {eng},
number = {2},
pages = {105-118},
title = {Extreme and exposed representing measures of the disk algebra},
url = {http://eudml.org/doc/262663},
volume = {73},
year = {2000},
}

TY - JOUR
AU - Heinis, Alex
AU - Wiegerinck, Jan
TI - Extreme and exposed representing measures of the disk algebra
JO - Annales Polonici Mathematici
PY - 2000
VL - 73
IS - 2
SP - 105
EP - 118
AB - We study the extreme and exposed points of the convex set consisting of representing measures of the disk algebra, supported in the closed unit disk. A boundary point of this set is shown to be extreme (and even exposed) if its support inside the open unit disk consists of two points that do not lie on the same radius of the disk. If its support inside the unit disk consists of 3 or more points, it is very seldom an extreme point. We also give a necessary condition for extreme points to be exposed and show that so-called BSZ-measures are never exposed.
LA - eng
KW - exposed point; representing measure; extreme point; disk algebra; exposed points; convex set; representing measures; BSZ-measures
UR - http://eudml.org/doc/262663
ER -

References

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  1. [1] L. Brown, A. Shields and K. Zeller, On absolutely convergent exponential sums, Trans. Amer. Math. Soc. 96 (1960), 162-183. Zbl0096.05103
  2. [2] R. Brummelhuis and J. Wiegerinck, Representing measures for the disc algebra and the ball algebra, Ann. Polon. Math. 55 (1991), 19-35. Zbl0765.30035
  3. [3] T. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. Zbl0213.40401
  4. [4] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. Zbl0469.30024
  5. [5] A. J. Izzo, Exposed points in the set of representing measures for the disc algebra, Ann. Polon. Math. 61 (1995), 59-62. Zbl0832.46047
  6. [6] P. Koosis, Theory of H p Spaces, Cambridge Univ. Press, Cambridge, 1980. 
  7. [7] J. Ryff, The support of representing measures for the disc algebra, in: Function Algebras, F. Birtel (ed.), Scott, Foresman and Co., Glenview, 1966, 112-117. Zbl0144.37603

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