Siciak’s homogeneous extremal functions, holomorphic extension and a generalization of Helgason’s support theorem
Volume 123 / 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.
