Definition:Little-Omega Notation
Definition
Informal Definition
Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.
$\omega$-notation is used to define a lower bound for $g$ which is not asymptotically tight.
Thus, let $f: \N \to \R$ be another real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.
Then:
- $\map f n = \map \omega {\map g n}$
means that $\map f n$ becomes arbitrarily large relative to $\map g n$ as $n$ approaches (positive) infinity.
Definition 1
Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.
Then $\map \omega g$ is defined as:
- $\map \omega g = \set {f: \N \to \R: \forall c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c \cdot \size {\map g n} < \size {\map f n} }$
Definition 2
Let $f: \N \to \R, g: \N \to \R$ be two real sequences, expressed here as real-valued functions on the set of natural numbers $\N$.
Let:
- $\ds \lim_{n \mathop \to \infty} {\frac {\map f n} {\map g n} } = \infty$
Then:
- $\map f n \in \map \omega {\map g n}$
Definition 3
Let $f: \N \to \R, g: \N \to \R$ be two real sequences, expressed here as real-valued functions on the set of natural numbers $\N$.
Then:
- $\map f n \in \map \omega {\map g n}$
- $\map g n \in \map \oo {\map f n}$
where $\oo$ denotes little-$\oo$ notation.
This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: Extract this into a separate page with proof You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page. |
A function $f$ is $\map \omega g$ if and only if $f$ is not $\map \OO g$ where $\OO$ is the big-$\OO$ notation.
Notation
The expression $\map f n \in \map \omega {\map g n}$ is read as:
- $\map f n$ is little-omega of $\map g n$
While it is correct and accurate to write:
- $\map f n \in \map \omega {\map g n}$
it is a common abuse of notation to write:
- $\map f n = \map \omega {\map g n}$
This notation offers some advantages.
Also known as
Note that in the Greek alphabet, $\omega$ is the minuscule counterpart of the capital letter $\Omega$.
Hence the former is called little-omega and the latter big-omega.
Some sources, therefore, write $\omega$ notation as little-$\omega$ notation, despite the fact that $\omega$'s "little"-ness is intrinsic.
$\mathsf{Pr} \infty \mathsf{fWiki}$ may sometimes adopt this convention if clarity is improved.
Also see
- Results about little-$\omega$ notation can be found here.