## Ragnar Sigurðsson (07/12/2018)

Valentina Giangreco, desember 3, 2018

Málstofa í stærðfræði

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

Staðsetning: VR-II 157
Tími: Föstudagur 7. desember kl. 11.40

### Ágrip:

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