  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.
