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Density of Sums of Shifts of a Single Function in Hardy Spaces on the Half-Plane

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Abstract

It is proved that there exists a function defined in the closed upper half-plane for which the sums of its real shifts are dense in all Hardy spaces Hp for 2 ≤ p < ∞, as well as in the space of functions analytic in the upper half-plane, continuous on its closure, and tending to zero at infinity.

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Acknowledgments

The author wishes to express gratitude to P. A. Borodin for posing the problem and useful remarks.

Funding

This work was supported by the Russian Foundation for Basic Research under grant 18-01-00333a.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    N. A. Dyuzhina

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  1. N. A. Dyuzhina
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Correspondence to N. A. Dyuzhina.

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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 5, pp. 669-678.

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Dyuzhina, N.A. Density of Sums of Shifts of a Single Function in Hardy Spaces on the Half-Plane. Math Notes 106, 711–719 (2019). https://doi.org/10.1134/S0001434619110051

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  • DOI: https://doi.org/10.1134/S0001434619110051

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