Approximations and expansions

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• ConceptApproximation theory

  Approximation theory studies how well complicated functions can be replaced by simpler ones, such as polynomials, splines or rational functions. It asks whether a best approximation exists, whether it is unique, and how to recognize and compute it. It also asks how fast the error of best approximation decreases. Its results give the error bounds behind numerical methods for computing functions, integrals and solutions of equations.

• ConceptFunction approximation

  Function approximation is the task of choosing, from a fixed class of simple functions, one that is close to a given target function. Closeness is measured by a norm, for instance the largest pointwise difference or a mean square difference. The task appears in numerical analysis, in signal processing and in machine learning, where the target is often known only through samples.

• ConceptConstructive function theory

  Constructive function theory is the part of approximation theory that describes a function through the speed of its approximation by polynomials. Its central results show that smoothness of $f$ and fast decay of the error of best approximation $E_n(f)$ go together. Sergei Bernstein introduced the name for this subject.

• ConceptBest approximation

  A best approximation of a point $x$ from a set $A$ in a normed space is a point of $A$ nearest to $x$. It is the optimal answer to an approximation problem. Existence, uniqueness and characterization of best approximations are the first questions of approximation theory.

• ConceptError of best approximation

  The error of best approximation of $x$ from a set $A$ is the distance from $x$ to $A$, the infimum of $\norm{x - a}$ over $a \in A$. For a function $f$ and the polynomials of degree at most $n$ it is written $E_n(f)$. It measures how well $f$ can be approximated at all, whether or not a best approximation exists. Its decay as $n \to \infty$ reflects the smoothness of $f$.

• ConceptUniform approximation

  Uniform approximation is function approximation in which the error is the largest pointwise difference, the supremum norm $\sup_x \abs{f(x) - g(x)}$. A function is uniformly approximable by a class when it is a uniform limit of functions from that class. It is the most demanding common way to measure an approximation, because it controls the error at every point at once.