On the best approximation of function classes from values on a uniform grid in the real line

  • Authors:

  • Gennadi Vainikko

  • Affiliations:

  • Institute of Applied Mathematics, Tartu University, Tartu, Estonia

  • Venue:
  • ICOSSE'06 Proceedings of the 5th WSEAS international conference on System science and simulation in engineering
  • Year:
  • 2006

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Abstract

Given the values of function f on a uniform grid of a step size h in the real line, we construct the spline interpolant of order m, defect 1 using the B-spline basis obtained by wavelet-type dilation and shifts from the father B-spline which can be considered as a scaling function for a special (nonorthogonal) spline-wavelet system. We establish an unimprovable error estimate on the class of functions with bounded mth derivative, and we show that in the sense of worst case, other approximations to f using the same information about f are more coarse than the spline interpolant. Some further results of this type are established.