Difference and functional equations Definition

Approximately additive function

Builds on Abelian group Normed vector space

Definition

Let $(G, +)$ be an abelian group, let $(E, \norm{\cdot})$ be a normed vector space, and let $\delta \ge 0$ . A map $f \colon G \to E$ is $\delta$ -additive if $\norm{f(x + y) - f(x) - f(y)} \le \delta \quad \text{for all } x, y \in G.$ A map is approximately additive if it is $\delta$ -additive for some $\delta \ge 0$ .

Remarks

The $0$ -additive maps are exactly the additive maps. Example: $f(x) = \floor{x}$ on $\R$ is $1$ -additive. Write $x = \floor{x} + s$ and $y = \floor{y} + t$ with $s, t \in [0, 1)$ ; then $\floor{x + y} - \floor{x} - \floor{y} = \floor{s + t} \in \set{0, 1}$ . Non-example: $f(x) = x^2$ on $\R$ is not approximately additive, since $f(x + y) - f(x) - f(y) = 2xy$ is unbounded. Every map $f = A + b$ with $A$ additive and $\norm{b(x)} \le \beta$ for all $x$ is $3\beta$ -additive.

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Abelian group concept
Normed vector space concept
Approximately additive function definition

Tag	`TP8XXM47CJ`
URI	`m2://TP8XXM47CJ`
Kind	definition
Writing	W0 stub
MSC	39B82, 39B52
Defines	Approximately additive function
Uses	Abelian group, Normed vector space

Approximately additive function

Definition

Remarks

Reading path

Further reading

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