Math 2.0

Approximations and expansions Definition

Constructive function theory

Builds on Approximation theoryError of best approximationModulus of continuity

Definition

Let ff be a function on [a,b][a, b], or a 2π2\pi-periodic function, in a fixed normed space. Let En(f)E_n(f) be the error of best approximation of ff by algebraic, respectively trigonometric, polynomials of degree at most nn. Constructive function theory is the part of approximation theory that characterizes classes of functions by the behaviour of the sequence (En(f))n≥0(E_n(f))_{n \ge 0} as n→∞n \to \infty. Its results are of two kinds.

  1. Direct theorems: a smoothness condition on ff, often a bound on its modulus of continuity, implies an upper bound for En(f)E_n(f).
  2. Inverse theorems: an upper bound for En(f)E_n(f) implies a smoothness condition on ff.

Remarks

Example: for 2π2\pi-periodic continuous ff, the uniform norm, and 0<α<10 < \alpha < 1, one has En(f)=O(n−α)E_n(f) = O(n^{-\alpha}) if and only if ∣f(x)−f(y)∣≤M∣x−y∣α\abs{f(x) - f(y)} \le M \abs{x - y}^\alpha for some constant MM and all x,yx, y. The direction from smoothness to the rate is a direct theorem of Jackson type, and the converse is an inverse theorem of Bernstein type.

Non-example: the statement that En(f)→0E_n(f) \to 0 for every continuous ff is a result of approximation theory. But it gives no rate and distinguishes no class of functions, so it is not of this kind.

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  1. Approximation theory concept
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Further reading

  1. Web https://en.wikipedia.org/wiki/Constructive_function_theory
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DefinesConstructive function theory
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