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

Journal article, 2019

The main result of the present paper is that a function defined on a union of lines CE through the origin in C-n with directional vectors in E subset of 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 CE 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.