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Fourier基函数、半周期余弦基函数和Chebyshev基函数张量积生成的标准正交基。我们研究与此基相关的高维逼近问题，并设计快速算法实现非等距Fourier矩阵、非等距余弦矩阵和非等距Chebyshev矩阵及其转置矩阵的行矩阵乘法。进而，通过在某些维上使用Fourier基，在其他维上使用半周期余弦基或Chebyshev基，得到了具有部分周期边界条件函数的ANOVA（方差分析）分解。在此框架下进行灵敏度分析，以寻找适应逼近问题的基。具体而言，确定多维级数展开的指标集合。此外，通过数值实验对混合基ANOVA逼近进行验证，并指出其可解释性结果的优势。