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Title: Reconstruction and collocation of a class of non-periodic functions by sampling along tent-transformed rank-1 lattices
Authors: Suryanarayana, Gowri ×
Nuyens, Dirk
Cools, Ronald #
Issue Date: 2016
Publisher: CRC Press
Series Title: The Journal of Fourier Analysis and Applications vol:22 pages:187-214
Abstract: Spectral collocation and reconstruction methods have been widely studied
for periodic functions using Fourier expansions.We investigate the use of cosine series
for the approximation and collocation of multivariate non-periodic functions with frequency
support mainly determined by hyperbolic crosses.We seek methods that work
for an arbitrary number of dimensions.We show that applying the tent-transformation
on rank-1 lattice points renders them suitable to be collocation/sampling points for the
approximation of non-periodic functions with perfect numerical stability. Moreover,
we show that the approximation degree—in the sense of approximating inner products
of basis functions up to a certain degree exactly—of the tent-transformed lattice
point set with respect to cosine series, is the same as the approximation degree of the
original lattice point set with respect to Fourier series, although the error can still be
reduced in the case of cosine series. A component-by-component algorithm is studied
to construct such a point set. We are then able to reconstruct a non-periodic function
from its samples and approximate the solutions to certain PDEs subject to Neumann
and Dirichlet boundary conditions. Finally, we present some numerical results.
ISSN: 1069-5869
Publication status: published
KU Leuven publication type: IT
Appears in Collections:NUMA, Numerical Analysis and Applied Mathematics Section
× corresponding author
# (joint) last author

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