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Mean Growth of Harmonic Functions of Beurling Type

Published online by Cambridge University Press:  20 November 2018

Manning G. Collier  and
John A. Kelingos
Manning G. Collier
Affiliation:
Mary Washington College, Fredericksburg, VA 22405, Vanderbilt University, Nashville TN 37235
John A. Kelingos
Affiliation:
Mary Washington College, Fredericksburg, VA 22405, Vanderbilt University, Nashville TN 37235
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Abstract

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A harmonic function on the unit disc is of Beurling type ω if its Fourier (or Taylor) coefficients grow no faster than exp ω(|n|) as |n|→∞, where ω is a given increasing, concave function with ω(x)/x ↓ 0 as x → ∞. These harmonic functions are characterized by the growth rate of their L1-norms on circles of radius r as r → 1. The classical Schwartz result follows as a corollary by taking ω(x) = log(1+x). The Gevrey case is also included in the general result if one uses ω(x) = xα, 0 < α < 1.

Keywords

31A0531A2546F05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 4 , 01 December 1984 , pp. 405 - 409
Copyright
Copyright © Canadian Mathematical Society 1984

References

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