Efficient multivariate approximation on the cube

Abstract

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 left[ -frac{1}{2},frac{1}{2}right] ^{d}. Our concept allows to determine conditions on the d-variate torus-to-cube transformations {psi :left[ -frac{1}{2},frac{1}{2}right] ^{d}rightarrow left[ -frac{1}{2},frac{1}{2}right] ^{d}} such that a non-periodic function is transformed into a smooth function in the Sobolev space {mathcal {H}}^{m}(mathbb {T}^{d}) when applying psi . We adapt L_{infty }(mathbb {T}^{d})- and L_{2}(mathbb {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.