Optimal recovery of isotropic classes of twice-differentiable multivariate functions

  • Authors:

  • Vladislav F. Babenko;Sergiy V. Borodachov;Dmytro S. Skorokhodov

  • Affiliations:

  • Dnepropetrovsk National University, vul. Naukova 13, Dnepropetrovsk, 49050, Ukraine and Institute of Applied Mathematics and Mechanics of NAS, Donetsk, 83114, Ukraine;Towson University, 8000 York Road, Towson, MD, 21252, USA;Dnepropetrovsk National University, vul. Naukova 13, Dnepropetrovsk, 49050, Ukraine

  • Venue:
  • Journal of Complexity
  • Year:
  • 2010

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Abstract

We consider the class of functions defined on a convex body in R^d, d@?N, whose second derivatives in any direction are uniformly bounded and the class of d-variate functions periodic with respect to a given full-rank lattice L and having uniformly bounded second derivative in any direction. The problem of the optimal algorithm which recovers functions from these classes using their values and values of their gradients at n points (nodes) is considered. We first obtain an estimate for the error of the optimal algorithms with fixed nodes. In the periodic case, for every n sufficiently large, we describe the optimal set of n nodes. When d=2, for certain periodic cases, optimality of the hexagonal arrangement of nodes is shown. For both the periodic case and the non-periodic case we present asymptotic results as n gets large.