Approximations and expansions Definition

Best approximation

Definition

Let $X$ be a normed vector space, let $A \subseteq X$ be nonempty, and let $x \in X$ . A point $a^* \in A$ is a best approximation of $x$ from $A$ if $\norm{x - a^*} \le \norm{x - a} \quad \text{for all } a \in A.$ Equivalently, $\norm{x - a^*} = \inf_{a \in A} \norm{x - a}$ , an infimum that is attained at $a^*$ .

Remarks

Example: in $\R^2$ with the Euclidean norm, let $A$ be the horizontal axis and $x = (1, 2)$ . For $a = (t, 0)$ one has $\norm{x - a}^2 = (1 - t)^2 + 4 \ge 4$ , with equality only at $t = 1$ , so $(1, 0)$ is the best approximation.

Non-example: in $\R$ , let $A = (0, 1]$ and $x = 0$ . The infimum of $\abs{0 - a}$ over $A$ is $0$ , but no $a \in A$ attains it, so $x$ has no best approximation from $A$ .

Other names are nearest point and, in the uniform norm, minimax approximation. The set of best approximations of $x$ from $A$ can be empty, a single point, or larger.

Reading path

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Normed vector space concept
Set bounded above definition
Infimum definition
Best approximation definition

Tag	`XEEZ2MQ6XM`
URI	`m2://XEEZ2MQ6XM`
Kind	definition
Writing	W0 stub
MSC	41A50, 41A65
Defines	Best approximation
Uses	Normed vector space, Infimum

Best approximation

Definition

Remarks

Reading path

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