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. explore the connection between periodic Fourier space non-periodic cosine Chebyshev space, via tent transform then transform, to transfer known results from setting into new insights settings. Fast discrete can be applied phase. To reduce size auxiliary set in associated component-by-component (CBC) construction lattice generating vectors, we work bi-orthonormal basis functions, leading three methods function provide theory efficient algorithmic strategies CBC construction. also interpret our context general approximation least-squares approximation.