# Extreme and exposed representing measures of the disk algebra

Annales Polonici Mathematici (2000)

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

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topHeinis, 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

top- [1] L. Brown, A. Shields and K. Zeller, On absolutely convergent exponential sums, Trans. Amer. Math. Soc. 96 (1960), 162-183. Zbl0096.05103
- [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] T. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. Zbl0213.40401
- [4] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. Zbl0469.30024
- [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] P. Koosis, Theory of ${H}^{p}$ Spaces, Cambridge Univ. Press, Cambridge, 1980.
- [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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