  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.
