FUNCTION INTEGRATION, RECONSTRUCTION AND APPROXIMATION USING RANK-1 LATTICES

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.

Automated summary

The study explores the connection between periodic Fourier space and non-periodic cosine space and Chebyshev space in the non-periodic settings, transferring known results using tent transform and cosine transform. Fast discrete cosine transform is applied for reconstruction, while a set of bi-orthogonal basis functions is used to reduce the size of the auxiliary index set in the component-by-component construction. New theory and efficient algorithmic strategies for CBC construction are provided, with results interpreted in the context of general function approximation and discrete least-squares approximation.