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.

我们将加权\（L_ {2} \）-积分的周期化策略与有效逼近方法结合在一起，以便在高维立方体\（\ left [-\ frac {1} {2} ，\ frac {1} {2} \ right] ^ {d} \）。我们的概念允许确定d个从圆环到多维数据集变换的条件\（{\ psi：\ left [-\ frac {1} {2}，\ frac {1} {2} \ right] ^ {d } \ rightarrow \ left [-\ frac {1} {2}，\ frac {1} {2} \ right] ^ {d}} \），这样非周期性函数在Sobolev中被转换为平滑函数应用\（\ psi \）时，请使用空格\（{\ mathcal {H}} ^ {m}（\ mathbb {T} ^ {d}）\）。我们改编\（L _ {\ infty}（\ mathbb {T} ^ {d}）\） -和\（L_ {2}（\ mathbb {T} ^ {d}）\） -单秩1格逼近方法的逼近误差估计和调整算法，用于快速评估和快速重建圆环上的多元三角多项式将这些方法应用于非定期设置。我们通过数值测试来说明高达\（d = 5 \）尺寸的理论发现。