In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. This leads us to an ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results.

翻译：本文考虑由傅里叶基函数、半周期余弦基函数和切比雪夫基函数的张量积生成的一组正交基。我们处理与此基相关的高维逼近问题，并设计一种快速算法，用于计算由非等距傅里叶矩阵、非等距余弦矩阵和非等距切比雪夫矩阵的行及其转置构成的矩阵乘法。由此，通过在某些维度使用傅里叶基，在其他维度使用半周期余弦基或切比雪夫基，我们得到部分周期性边界条件下函数的ANOVA（方差分析）分解。在此框架下，我们进行灵敏度分析，以寻找适用于底层逼近问题的自适应基。更精确地，我们确定多维级数展开的底层指标集。此外，我们通过数值实验测试这种混合基的ANOVA逼近方法，并指出其结果可解释性的优势。