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.

中文翻译：

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使用秩 1 格的函数积分、重构和逼近

我们考虑 rank-1 格子，用于在有限索引集上支持级数扩展的函数的积分和重构。我们通过帐篷变换和余弦变换探索周期性傅立叶空间与非周期性余弦空间和切比雪夫空间之间的联系，将已知结果从周期性设置转换为非周期性设置的新见解。快速离散余弦变换可以应用于重建阶段。为了减少格生成向量的相关组件逐组件 (CBC) 构造中辅助索引集的大小，我们使用双正交基函数集，导致在非定期设置。我们为 CBC 构建提供了新的理论和有效的算法策略。