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Functions Invariant under the Berezin Transform

✍ Scribed by M. Englis

Publisher
Elsevier Science
Year
1994
Tongue
English
Weight
587 KB
Volume
121
Category
Article
ISSN
0022-1236

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✦ Synopsis

We show that a bounded function (f) satisfies (B f=f), where (B) is the Berezin tranform on the unit disc (defined in (2) below), if and only if (f) is harmonic. There is an equivalent formulation of this result [S. Axler and Z. CučkoviΔ‡, Integral Equations Operator Theory 14 (1991), 1-12; W. Rudin, "Function Theory in the Unit Ball of (C^{N})," Springer-Verlag, New York/Berlin, 1980]: If (f) is bounded and satifies the invariant version of the area mean value property, then (f) is harmonic. The main tool employed is Fourier analysis on the Lie group of all MΓΆbius transformations. (C) 1994 Academic Press, Inc.

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