Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T08:58:29.394Z Has data issue: false hasContentIssue false
  • English
  • Français

Some dual aspects of the Poisson kernel

Published online by Cambridge University Press:  20 January 2009

F. F. Bonsall
F. F. Bonsall
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, England
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 2 , June 1990 , pp. 207 - 232
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

REFERENCES

1.Bonsall, F. F., Decomposition of functions as sums of elementary functions, Quart. J. Math. Oxford (2), 37 (1986), 129–136.CrossRefGoogle Scholar
2.Bonsall, F. F., Domination of the supremum of a bounded harmonic function by its supremum over a countable subset, Proc. Edinburgh Math. Soc. 30 (1987), 471–477.CrossRefGoogle Scholar
3.Bonsall, F. F. and Walsh, D., Vanishing l 1-sums of the Poisson kernel, and sums with positive coefficients, Proc. Edinburgh Math. Soc., to appear.Google Scholar
4.Dunford, N. and Schwartz, J. T., Linear Operators (New York, 1958).Google Scholar
5.Garnett, J. B., Interpolating sequences for bounded harmonic functions, Indiana Univ. Math. J. 21 (1971), 187–192.CrossRefGoogle Scholar
6.Garnett, J. B., Bounded analytic functions (New York, 1981).Google Scholar
7.Hayman, W. K. and Lyons, T. J., Bases for positive continuous functions, to appear.Google Scholar
8.Newman, M. H. A., Topology of plane sets of points (Cambridge, 1939).Google Scholar
9.Rudin, W., Real and complex analysis (New York, 1966).Google Scholar
10.Sarason, D., Function theory on the unit circle, Lecture Notes (Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1978).Google Scholar
11.Shapiro, H. S. and Shields, A. L., On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532.CrossRefGoogle Scholar