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

## TL;DR

This paper establishes conditions under which functions defined on complex lines extend to entire functions in multiple variables, linking Siciak's extremal functions with a generalized Helgason support theorem.

## Contribution

It generalizes Helgason's support theorem by connecting holomorphic extension criteria with Siciak's extremal functions and capacity conditions.

## Key findings

- Functions on complex lines extend holomorphically under capacity and derivative conditions.
- The indicator function of the extension is estimated via Siciak's extremal function.
- A generalized support theorem relates Radon transform support to original function support.

## Abstract

We prove that a function, which is defined on a union of lines $\mathbb{C} E$ through the origin in $\mathbb{C}^n$ with direction vectors in $E\subset \mathbb{C}^n$ and is holomorphic of fixed finite order and finite type along each line, extends to an entire holomorphic function on $\mathbb{C}^n$ of the same order and finite type, provided that $E$ has positive homogeneous capacity in the sense of Siciak 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 on the support of its Radon transform.