Abstract
The approximation of problems in \(d\) spatial dimensions by trigonometric polynomials supported on known more or less sparse frequency index sets \(I\subset \mathbb {Z}^d\) is an important task with a variety of applications. The use of rank-1 lattices as spatial discretizations offers a suitable possibility for sampling such sparse trigonometric polynomials. Given an arbitrary index set of frequencies, we construct rank-1 lattices that allow a stable and unique discrete Fourier transform. We use a component-by-component method in order to determine the generating vector and the lattice size.
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References
Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)
Cooley, J.W., Tukey, J.W.: An algorithm for machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)
Cools, R., Kuo, F.Y., Nuyens, D.: Constructing lattice rules based on weighted degree of exactness and worst case error. Computing 87, 63–89 (2010)
Kämmerer, L.: Reconstructing hyperbolic cross trigonometric polynomials by sampling along rank-1 lattices. SIAM J. Numer. Anal. 51, 2773–2796 (2013). http://dx.doi.org/10.1137/120871183
Kühn, T., Sickel, W., Ullrich, T.: Approximation numbers of Sobolev embeddings—sharp constants and tractability. J. Complex. 30, 95–116 (2013). doi:10.1016/j.jco.2013.07.001
Li, D., Hickernell, F.J.: Trigonometric spectral collocation methods on lattices. In: Cheng, S.Y., Shu, C.-W., Tang T. (eds.) Recent Advances in Scientific Computing and Partial Differential Equations, AMS Series in Contemporary Mathematics, vol. 330, pp. 121–132. American Mathematical Society, Providence, RI (2003)
Munthe-Kaas, H., Sørevik, T.: Multidimensional pseudo-spectral methods on lattice grids. Appl. Numer. Math. 62, 155–165 (2012). doi:10.1016/j.apnum.2011.11.002
Sickel, W., Ullrich, T.: The Smolyak algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness. East J. Approx. 13, 387–425 (2007)
Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford Science Publications, The Clarendon Press Oxford University Press, New York (1994)
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The author thanks the referees for their careful reading and their valuable suggestions for improvements.
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Kämmerer, L. (2014). Reconstructing Multivariate Trigonometric Polynomials from Samples Along Rank-1 Lattices. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_14
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DOI: https://doi.org/10.1007/978-3-319-06404-8_14
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