Sie befinden Sich nicht im Netzwerk der Universität Paderborn. Der Zugriff auf elektronische Ressourcen ist gegebenenfalls nur via VPN oder Shibboleth (DFN-AAI) möglich. mehr Informationen...
Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums
Ist Teil von
The Journal of fourier analysis and applications, 2019-08, Vol.25 (4), p.1580-1608
Ort / Verlag
New York: Springer US
Erscheinungsjahr
2019
Link zum Volltext
Quelle
Alma/SFX Local Collection
Beschreibungen/Notizen
In this paper, we study the convergence of adaptive Fourier sums for real-valued
2
π
-periodic functions. For this purpose, we approximate the sequence of classical Fourier coefficients by a short exponential sum with a pre-defined number of
N
+
1
terms. The obtained approximation can be interpreted as an adaptive
N
-th Fourier sum with respect to the orthogonal Takenaka-Malmquist basis. Using the theoretical results on rational approximation in Hardy spaces and on the decay of singular values of special infinite Hankel matrices, we show that adaptive Fourier sums can converge essentially faster than classical Fourier sums for a large class of functions. Further, we derive an algorithm to compute almost optimal adaptive Fourier sums. Our numerical results show that the significantly better convergence behavior of adaptive Fourier sums for optimally chosen basis elements can also be achieved in practice. For comparison, we also provide a greedy algorithm to determine an adaptive Fourier sum. This algorithm requires less computational effort but yields essentially slower convergence.