## Abstract

We propose isotropic probability measures defined on classes of smooth multivariate functions. These provide a natural extension of the classical isotropic Wiener measure to multivariate functions from C^{2r}. We show that, in the corresponding average case setting, the minimal errors of algorithms that use n function values are Θ(n^{−(d+4r+1)/(2d)}) and Θ(n^{−(4r+1)/(2d)}) for the integration and L_{2}-approximation problems, respectively. Here d is the number of variables of the corresponding class of functions. This means that the minimal average errors depend essentially on the number d of variables. In particular, for d large relative to r, the L_{2}-approximation problem is intractable. The integration and L_{2}-approximation problems have been recently studied with measures whose covariance kernels are tensor products. The results for these measures and for isotropic measures differ significantly.

Original language | English |
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Pages (from-to) | 1541-1557 |

Number of pages | 17 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - 1996 |

## ASJC Scopus subject areas

- General Mathematics

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