Cubature and reconstruction of smooth isotropic random functions

K. Ritter, G. W. Wasilkowski

Research output: Contribution to journalArticlepeer-review


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 C2r. 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 L2-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 L2-approximation problem is intractable. The integration and L2-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 languageEnglish
Pages (from-to)124-127
Number of pages4
JournalZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
Issue numberSUPPL. 3
StatePublished - 1996

ASJC Scopus subject areas

  • Computational Mechanics
  • Applied Mathematics


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