Efficient multivariate approximation on the cube

We combine a periodization strategy for weighted L-2-integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube [-1/2, 1/2](d). Our concept allows to determine conditions on the d-variate torus-to-cube transformations psi : [-1/2, 1/2](d) -> [-1/2, 1/2](d) such that a nonperiodic function is transformed into a smooth function in the Sobolev space H-m(T-d) when applying psi. We adapt L-infinity(T-d)- and L-2(T-d)-approximation error estimates for single rank-1 lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to d = 5 dimensions.