Numerical Function Approximation Task

A Numerical Function Approximation Task is an Function Approximation Task required to produce an Approximation Function for a given Numerical Function.

• See: Numerical Function Approximation Theory, Taylor Series.

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References

2007

• (de Boor et al., 2007) ⇒ Carl de Boor, Allan Pinkus, Vilmos Totik. (2007). “Concepts of Approximation Theory.” In: Surveys in Approximation Theory Web Site, July 21, 2007.
  • approximand is the element to be approximated.
  • approximant is the element that is doing the approximating.
  • approximation spaces consist of functions with prescribed rate of approximation. E.g. if En(f) is the error of best approximation of f by polynomials of degree at most [math]\displaystyle{ n }[/math], [math]\displaystyle{ α }[/math] > 0 and 0 < q ≤ ∞, then the collection of functions f for which ...
  • approximation with constraint: An additional requirement (like monotonicity, convexity, interpolation) has to be satisfied by the approximation process.
  • best approximant to f from M in a metric space X (with metric ρ) containing M is any element g ∈ M for which ρ(f, g) is minimal, i.e., ρ(f, g) = infh∈M ρ(f, h) =: dist(f,M). Also called best approximation.
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