Function integration, reconstruction and approximation using rank-1 lattices
Abstract
We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.
Keywords
Cite
@article{arxiv.1908.01178,
title = {Function integration, reconstruction and approximation using rank-1 lattices},
author = {Frances Y. Kuo and Giovanni Migliorati and Fabio Nobile and Dirk Nuyens},
journal= {arXiv preprint arXiv:1908.01178},
year = {2020}
}