Efficient function reconstruction using eigenfunctions of linear operators
(07/201406/2017)

Individual research grant of the German Research Foundation
Goal of this project is the generalization of nonlinear reconstruction methods that are based on Prony's method. Here we apply a new view on this approach, namely that the classical Prony method for parameter identification in exponential sums as well as the BenOr and Tiwari algorithm for sparse polynomial interpolation can be understood as nonlinear reconstruction techniques for Mterm expansions of eigenfunctions of special linear operators. We are particularly interested in deriving fast and numerically stable reconstruction schemes as well as new error estimates in case of noisy input data. 
Application of the AAK theory for sparse approximation of exponential sums Gerlind Plonka, Vlada Pototskaia University of Göttingen, Institute for Numerical and Applied Mathematics, September 2016, preprint as download.  
A sparse Fast Fourier algorithm for real nonnegative vectors Gerlind Plonka, Katrin Wannenwetsch Journal of Computational and Applied Mathematics 321, 532539., 2017, preprint as download.  
Deterministic Sparse FFT Algorithms Katrin Wannenwetsch Dissertation, 2016, published online on 30 September 2016, http://hdl.handle.net/11858/0017350000002B7C100.  
Sparse approximation by Prony's method and AAK theory Gerlind Plonka, Vlada Pototskaia Oberwolfach Reports, Volume 33, pp. 1619, 2016, preprint as download.  
Reconstruction of polygonal shapes from sparse Fourier samples Marius Wischerhoff, Gerlind Plonka Journal of Computational and Applied Mathematics 297, 117131, 2016, preprint as download.  
A deterministic sparse FFT algorithm for vectors with small support Gerlind Plonka, Katrin Wannenwetsch Numerical Algorithms 71(4), 889905, 2016, preprint as download.  
Deterministic sparse FFT algorithms Gerlind Plonka, Katrin Wannenwetsch Proc. Appl. Math. Mech. Volume 15, Issue 1, pp. 667668, October 2015, DOI: 10.1002/pamm.201510323, preprint as download.  
Prony's Method for Multivariate Signals Thomas Peter, Gerlind Plonka, Robert Schaback Proc. Appl. Math. Mech. Volume 15, Issue 1, pp. 665666, October 2015, DOI: 10.1002/pamm.201510322, preprint as download.  
A deterministic sparse FFT algorithm for vectors with short support Gerlind Plonka, Katrin Wannenwetsch Oberwolfach Reports, Volume 38, pp. 4144, 2015,  
Prony methods for recovery of structured functions Gerlind Plonka, M. Tasche GAMMMitt. 37(2), 2014, 239258, revised preprint as download. 
Sparse FFT (small support) A new deterministic sparse FFT algorithm for vectors with small support. Katrin Wannenwetsch, Gerlind Plonka  
Sparse FFT (real nonnegative) A new deterministic sparse FFT algorithm for real nonnegative vectors. Katrin Wannenwetsch, Gerlind Plonka 