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Interpolation Operators as Optimal Recovery Schemes for Classes of Analytic Functions

  • Michael Golomb
Part of the The IBM Research Symposia Series book series (IRSS)

Abstract

Suppose that the only information we have about a function f is that it belongs to a certain class ℬ (usually a ball in a normed space) and that it takes on given values at some finitely many points x1,...,xn of its domain (or that some other finitely many linear functionals ℓ1,..,ℓn have given values at f). Suppose we are to assign a value to f(x) where x does not belong to the set {x1,...,xn} (or to ℓ(f) where ℓ is not in the span of {ℓ1,...,ℓn}). The value α* for f(x) is considered optimal if for any other assignment α there is some fα ∈ ℬ with fα(xi) = f(xi) (i = l,...,n) for which the error |α − fα(x)| is at least as large as |α* − f(x)|. If moreover a function s* ∈ ℬ can be found such that s* evaluated at x gives the optimal value α* and this is so for every x in the domain of the functions f then s* is considered an optimal interpolant (or extrapolant) for these functions.

Keywords

Hilbert Space Analytic Continuation Minimal Error Simple Polis Recovery Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Michael Golomb
    • 1
  1. 1.Purdue UniversityWest LafayetteUSA

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