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Siciak’s homogeneous extremal functions, holomorphic extension and a generalization of Helgason’s support theorem

Volume 123 / 2019

Jöran Bergh, Ragnar Sigurdsson Annales Polonici Mathematici 123 (2019), 61-70 MSC: Primary 32A15; Secondary 30D15, 32U35. DOI: 10.4064/ap190128-22-7 Published online: 26 September 2019

Abstract

The main result of the present paper is that a function defined on a union of lines $\mathbb C E$ through the origin in $\mathbb C ^n$ with directional vectors in $E \subset \mathbb C ^n$ and holomorphic of fixed finite order and finite type along each of these lines can be extended to an entire function of the same order and finite type provided that $\mathbb C E$ is not pluripolar and all directional derivatives along the lines satisfy a necessary compatibility condition at the origin. We are able to estimate the indicator function of the extension in terms of Siciak’s weighted homogeneous extremal function, where the weight is a function of the type of the given function on each given line. As an application we prove a generalization of Helgason’s support theorem by showing how the support of a continuous function with rapid decrease at infinity can be located from partial information about the support of its Radon transform.

Authors

  • Jöran BerghDepartment of Mathematical Sciences
    Chalmers University of Technology and
    University of Gothenburg
    SE-412 96 Göteborg, Sweden
    e-mail
  • Ragnar SigurdssonDepartment of Mathematics
    School of Engineering and Natural Sciences
    University of Iceland
    IS-107 Reykjavík, Iceland
    e-mail

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