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2 results

  • Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows

  • Xiang-Jun Zhou, Lei Shi, Ding-Xuan Zhou
  • Journal:
    Canadian Mathematical Bulletin / Volume 54 / Issue 3 / 01 September 2011
    Published online by Cambridge University Press:
    20 November 2018, pp. 566-576
    Print publication:
    01 September 2011
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  • We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.

  • Tubular sets and multivariate Polya algorithm

  • Part of
  • Robert Huotari
  • Journal:
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 55 / Issue 2 / October 1993
    Published online by Cambridge University Press:
    09 April 2009, pp. 232-237
    Print publication:
    October 1993
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  • Some new results concerning tubular sets are presented, with applications to the convergence of the Polya algorithm in the contexts of simultaneous approximation and approximation of multivariate functions by univariate functions. (The Polya algorithm constructs a best uniform approximation from the limit, as p → ∞, of best Lp approximations.)