## Ragnar Sigurðsson (07/12/2018)

### Speaker: Ragnar Sigurðsson, University of Iceland

### Title: Siciak’s extremal functions and Helgason’s support theorem

Location: VR-II 157

Time: Friday December 7 at 11.40

### Abstract:

We prove that a function, which is defined on a union

of lines $\C E$ through the origin in $\C^n$ with direction

vectors in $E\subset \C^n$ and is holomorphic

of fixed finite order and finite type along each line,

extends to an entire holomorphic function on $\C^n$

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 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.

This is a joint work with Jöran Bergh at Chalmers University of

Technology and University of Gothenburg.