Sampling sets for Hardy spaces of the disk

Thomas, Pascal

doi:10.1090/S0002-9939-98-04411-6

Sampling sets for Hardy spaces of the disk
HTML articles powered by AMS MathViewer

by Pascal J. Thomas PDF

Proc. Amer. Math. Soc. 126 (1998), 2927-2932 Request permission

Abstract:

We propose two possible definitions for the notion of a sampling sequence (or set) for Hardy spaces of the disk. The first one is inspired by recent work of Bruna, Nicolau, and Øyma about interpolating sequences in the same spaces, and it yields sampling sets which do not depend on the value of $p$ and correspond to the result proved for bounded functions ($p=\infty$) by Brown, Shields and Zeller. The second notion, while formally closer to the one used for weighted Bergman spaces, leads to trivial situations only, but raises a possibly interesting problem.

References

Joaquim Bruna, Artur Nicolau, and Knut Øyma, A note on interpolation in the Hardy spaces of the unit disc, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1197–1204. MR 1307499, DOI 10.1090/S0002-9939-96-03168-1
Leon Brown, Allen Shields, and Karl Zeller, On absolutely convergent exponential sums, Trans. Amer. Math. Soc. 96 (1960), 162–183. MR 142763, DOI 10.1090/S0002-9947-1960-0142763-8
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37–52. MR 222554, DOI 10.1007/BF02395039
Daniel H. Luecking, Dominating measures for spaces of analytic functions, Illinois J. Math. 32 (1988), no. 1, 23–39. MR 921348
Yurii I. Lyubarskii and Kristian Seip, A uniqueness theorem for bounded analytic functions, Bull. London Math. Soc. 29 (1997), no. 1, 49–52. MR 1416406, DOI 10.1112/S0024609396001816
Kristian Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), no. 1, 21–39. MR 1223222, DOI 10.1007/BF01244300
Kristian Seip, Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc. 117 (1993), no. 1, 213–220. MR 1111222, DOI 10.1090/S0002-9939-1993-1111222-5
H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532. MR 133446, DOI 10.2307/2372892
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095