Approximations and expansions Definition

Error of best approximation

Builds on Normed vector space Infimum Best approximation Polynomial function

Definition

Let $X$ be a normed vector space, let $A \subseteq X$ be nonempty, and let $x \in X$ . The error of best approximation of $x$ from $A$ is the infimum $E(x, A) = \inf_{a \in A} \norm{x - a}.$ In particular, for a continuous $f \colon [a, b] \to \R$ with the supremum norm and the set $\mathcal{P}_n$ of real polynomial functions of degree at most $n$ , $E_n(f) = \inf_{p \in \mathcal{P}_n} \max_{a \le x \le b} \abs{f(x) - p(x)}.$

Remarks

Example: for $f(x) = x^2$ on $[0, 1]$ one has $E_1(f) = 1/8$ . The line $p(x) = x - 1/8$ gives $\abs{x^2 - x + 1/8} \le 1/8$ on $[0, 1]$ , and no line does better, because the error of $p$ takes the values $1/8, -1/8, 1/8$ at $0, 1/2, 1$ .

Non-example: the error of a particular approximation is in general larger. The line $x$ has error $\max_{[0, 1]} \abs{x^2 - x} = 1/4$ , which is not $E_1(f)$ .

$E(x, A) = 0$ if and only if $x$ lies in the closure of $A$ . The infimum equals $\norm{x - a^*}$ when $a^*$ is a best approximation. Since $\mathcal{P}_0 \subseteq \mathcal{P}_1 \subseteq \cdots$ , the sequence $E_n(f)$ is nonincreasing.

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Tag	`C0AKF92CWH`
URI	`m2://C0AKF92CWH`
Kind	definition
Writing	W0 stub
MSC	41A50, 41A25, 41A10
Defines	Error of best approximation
Uses	Normed vector space, Infimum, Best approximation, Polynomial function

Error of best approximation

Definition

Remarks

Reading path

Further reading

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