Definition:Big-O Notation

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Definition

Sequences

Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.

Then $\map \OO g$ is defined as:

$\map \OO g = \set {f: \N \to \R: \exists c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le \size {\map f n} \le c \cdot \size {\map g n} }$


Real Analysis

Let $f$ and $g$ be real functions defined on a neighborhood of $+ \infty$ in $\R$.


The statement:

$\map f x = \map \OO {\map g x}$ as $x \to \infty$

is equivalent to:

$\exists c \in \R_{\ge 0}: \exists x_0 \in \R: \forall x \in \R: \paren {x \ge x_0 \implies \size {\map f x} \le c \cdot \size {\map g x} }$


That is:

$\size {\map f x} \le c \cdot \size {\map g x}$

for $x$ sufficiently large.


This statement is voiced $f$ is big-$\OO$ of $g$ or simply $f$ is big-$\OO$ $g$.


Complex Analysis

Let $f$ and $g$ be complex functions defined for all complex numbers whose modulus is sufficiently large.


The statement:

$\map f z = \map \OO {\map g z}$ as $\cmod z \to \infty$

is equivalent to:

$\exists c \in \R_{\ge 0}: \exists r_0 \in \R: \forall z \in \C: \paren {\cmod z \ge r_0 \implies \cmod {\map f z} \le c \cdot \cmod {\map g z} }$


That is:

$\cmod {\map f z} \le c \cdot \cmod {\map g z}$

for all $z$ in a neighborhood of infinity in $\C$.


General Definition

Let $\struct {X, \tau}$ be a topological space.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$.

Let $f, g : X \to V$ be functions.


The statement:

$\map f x = \map \OO {\map g x}$ as $x \to \infty$

is equivalent to:

There exists a neighborhood of infinity $U \subset X$ such that:
$\exists c \in {\R}_{\ge 0}: \forall x \in U: \norm {\map f x} \le c \norm {\map g x}$


That is:

$\norm {\map f x} \le c \norm {\map g x}$

for all $x$ in a neighborhood of infinity.


Uniform Estimates

Let $X$ be a set.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\, \cdot \,}$.

Let $f, g : X \to V$ be mappings.


Then $f$ is big-$\OO$ of $g$ uniformly if and only if:

$\exists c > 0 : \forall x \in X : \norm {\map f x} \le c \cdot \norm {\map g x}$

This is denoted:

$f = \map \OO g$


Implied Constant

From the definition of the limit of a function, it can be seen that this is also equivalent to:

$\exists c \in \R: c > 0, k \ge 0: \forall n > k: \map f n \le c \map g n$

For some fixed $k$ (appropriate to the function under consideration) the infimum of such $c$ is called the implied constant.


Also known as

The big-$\OO$ notation, along with little-$\oo$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.


Also denoted as

In analytic number theory, sometimes Vinogradov's notations $f \ll g$ or $g \gg f$ are used for big-$\OO$ notation, to mean $f = \map \OO g$.

This can often be clearer for estimates leading to typographically complex error terms.


Some sources use an ordinary $O$:

$f = \map O g$


Also defined as

Some authors require that the inequality be valid on the entire domain of definition.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, this is known as a uniform big-$\OO$ estimate.


The statement $f = \map \OO g$ is sometimes seen to be defined as:

$\ds \exists \alpha \in \R_{\ge 0}: \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = \alpha$

However, requiring that the limit exists is generally viewed to be too restrictive.


Also see

  • Results about big-$\OO$ notation can be found here.


Sources

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