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Vanishing l1-sums of the Poisson kernel, and sums with positive coefficients

  • F. F. Bonsall (a1) and D. Walsh (a2)
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For z in D and ζ in ∂D, we denote by pz(ζ) the Poisson kernel (1 − │z│2)│1 − ζ−2 for the open unit disc D. We ask for what countable sets {an:n∈ℕ} of points of D there exist complex numbers λn with

by which we mean that the series converges to zero in the norm of L1(∂D).

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Copyright
COPYRIGHT: © Edinburgh Mathematical Society 1989
References
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1.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.
2.Brown, L., Shields, A. and Zeller, K., On absolutely convergent exponential sums, Trans. American Math. Soc. 96 (1960), 162–183.
3.Dunford, N. and Schwartz, J. T., Linear Operators (New York, 1958).
4.Garnett, J., Interpolating sequences for bounded harmonic functions, Indiana Univ. Math. J. 21 (1971), 187–192.
5.Hayman, W. K. and Lyons, T. J., Bases for positive continuous functions, to appear.
6.Saks, S., Theory of the Integral (Warsaw, 1937).
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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