# Function integration, reconstruction and approximation using rank-1 lattices

## TL;DR

This paper develops new methods for function integration, reconstruction, and approximation using rank-1 lattices, leveraging transformations between periodic and non-periodic function spaces to improve efficiency and accuracy.

## Contribution

It introduces novel theoretical insights and algorithms for rank-1 lattice-based function reconstruction in non-periodic settings, including bi-orthonormal basis methods and CBC construction strategies.

## Key findings

- Fast discrete cosine transform enables efficient reconstruction.
- New bounds and algorithms for CBC construction reduce auxiliary set size.
- Methods improve accuracy and efficiency in non-periodic function approximation.

## 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.