title:
Reconstructing multivariate trigonometric polynomials by sampling along generated sets 
name:
Kaemmerer 
first name:
Lutz

location/conference:
SPPJT12

PRESENTATIONlink:
http://www.dfgspp1324.de/nuhagtools/event_NEW/dateien/SPPJT12/talks/Kaemmererjt12.pdf 
abstract:
The approximation of problems in $d$ spatial dimensions by sparse trigonometric polynomials supported on known or unknown frequency sets $I\subset\mathbb{Z}^d$ is an important task with a variety of applications.
The use of a generalisation of rank1 lattices as spatial discretisations offers a suitable possibility for sampling such sparse trigonometric polynomials.
We assume the knowledge of the frequency set $I\subset\mathbb{Z}^d$ and construct corresponding sampling sets that allow a stable and unique discrete Fourier transform.
A multivariate trigonometric polynomial evaluated at all nodes of these sampling sets essentially simplifies to a onedimensional nonequispaced discrete Fourier transform of the length of the number of used sampling nodes.
Applying the onedimensional nonequispaced fast Fourier transform enables the fast evaluation and reconstruction of the multivariate trigonometric polynomials. 