Málstofa í stærðfræði Í DAG 10. september: Ragnar Sigurðsson

[Aron Lagerberg]

Dagsetning: Mánudagur, 10 september, 15.00

Staður: VRII herbergi 157

Fyrirlesari: Ragnar Sigurðsson, University of Iceland

Titill: Growth estimates of entire functions on $\C^n$ of finite
order and type along certain complex lines}

Ágrip:
This lecture is report on a joint work with J\"oran Bergh in Gothenburg.
We prove that an entire function on $\C^n$ which is of finite order
and type along a set of lines through the origin with direction
vectors in a non-pluripolar set $E$ is of the same order in the whole
space and we estimate its radial indicator function in terms of
Siciak's homogeneous extremal function for the set $E$.
By using an explicit formula for Siciac's function for the circular hull
of the closed unit ball in $\R^n$ we are able describe its polynomial
hull and conclude that every entire function which is of exponential type
$\sigma$ along $\C\R^n$ is of exponential type $\leq \sqrt 2\, \sigma$
in the whole space. Furthermore, this enables us relax conditions in
the Paley-Wiener theorem by only assuming that a function is entire and
of exponential type in non-pluripolar circular set of directions
and of polynomial growth in real directions in order to conclude
that it is the Fourier-Laplace transform of a distribution with
compact support.