Math 2.0

Approximations and expansions Definition

Approximation theory

Builds on Normed vector spaceFunction approximationBest approximationError of best approximation

Definition

Let XX be a normed vector space of functions, and let A0⊆A1⊆A2⊆⋯A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots be nonempty subsets of XX, the approximating classes. Approximation theory is the study of the following questions for f∈Xf \in X.

  1. Existence: does ff have a best approximation from AnA_n?
  2. Uniqueness and characterization: is it unique, and which conditions single it out?
  3. Rates: how fast does the error of best approximation E(f,An)E(f, A_n) tend to 00 as n→∞n \to \infty, and how does this rate depend on ff?
  4. Construction: which explicit methods give good approximations of ff from AnA_n?

Remarks

Example: X=C[a,b]X = C[a, b] with the supremum norm and AnA_n the polynomials of degree at most nn. Here a best approximation exists and is unique, it is characterized by the equioscillation theorem, and E(f,An)→0E(f, A_n) \to 0 for every ff by the Weierstrass approximation theorem.

Non-example: a single numerical estimate, such as ∣ex−(1+x)∣≤1\abs{e^x - (1 + x)} \le 1 on [0,1][0, 1], is an approximation result but does not by itself pose these questions. The theory studies whole families AnA_n and all ff in a class.

Other common choices are trigonometric polynomials, splines with nn knots, rational functions of degree nn, and LpL^p norms instead of the supremum norm.

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Best approximation, concept, no evidence levelBest approximationconceptError of best approximation, concept, no evidence levelError of best approxi…conceptFunction approximation, concept, no evidence levelFunction approximationconceptNormed vector space, concept, no evidence levelNormed vector spaceconceptApproximation theory, definition, no evidence levelApproximation theorydefinition
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  1. Best approximation concept
  2. Error of best approximation concept
  3. Function approximation concept
  4. Normed vector space concept
  5. Approximation theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Approximation_theory
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MSC41A65, 41A10, 41A50
DefinesApproximation theory
UsesNormed vector space, Function approximation, Best approximation, Error of best approximation

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