## Abstract

We present formulas that allow us to decompose a function f of d variables into a sum of 2d terms fu indexed by subsets u of {1,..., d}, where each term fu depends only on the variables with indices in u. The decomposition depends on the choice of d commuting projections {Pj}^{d}_{j=1}where Pj (f) does not depend on the variable x_{j}. We present an explicit formula for f_{u}, which is new even for the anova and anchored decompositions; both are special cases of the general decomposition. We show that the decomposition is minimal in the following sense: if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z, then, for every choice of {Pj}^{d}_{j=1} the terms fu = 0 for all subsets u containing z. Furthermore, in a reproducing kernel Hilbert space setting, we give sufficient conditions for the terms f_{u} to be mutually orthogonal.

Original language | English |
---|---|

Pages (from-to) | 953-966 |

Number of pages | 14 |

Journal | Mathematics of Computation |

Volume | 79 |

Issue number | 270 |

DOIs | |

State | Published - Apr 2010 |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics