Efficient multivariate approximation on the cube

For the approximation of multivariate non-periodic functions $h$ on the high-dimensional cube $\left[-\frac{1}{2},\frac{1}{2}\right]^{d}$ we combine a periodization strategy for weighted $L_{2}$-integrands with efficient approximation methods. We prove sufficient conditions on $d$-variate torus-to-cube transformations ${\psi:\left[-\frac{1}{2},\frac{1}{2}\right]^{d}\to\left[-\frac{1}{2},\frac{1}{2}\right]^{d}}$ and on the non-negative weight function $\omega$ such that the composition of a possibly non-periodic function with a transformation $\psi$ yields a smooth function in the Sobolev space $H_{\mathrm{mix}}^{m}(\mathbb{T}^{d})$. In this framework we adapt certain $L_{\infty}(\mathbb{T}^{d})$- and $L_{2}(\mathbb{T}^{d})$-approximation error estimates for single rank-$1$ lattice approximation methods as well as algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus to the non-periodic setting. Various numerical tests in up to dimension $d=5$ confirm the obtained theoretical results for the transformed approximation methods.